BL-rings

BL-rings Abstract The main goal of this article is to initiate the study of commutative rings whose lattice of ideals can be equipped with a structure of BL-algebra, referred to henceforth as BL-rings. We obtain a description of such rings in terms of their subdirectly irreducible factors. We also investigate their connections with some prominent classes of commutative rings such as multiplication rings, Baer rings, Dedekind rings and Von Neumann rings. 1 Introduction Given any ring R (commutative or not, with or without unity) generated by idempotents, the semiring of ideals of R under the usual operations form a residuated lattice A(R). In recent articles, several authors have investigated classes of rings for which the residuated lattice A(R) is an algebra of a well-known subvariety of residuated lattices. For instance, rings R for which A(R) is an MV-algebra, also called Łukasiewicz rings are investigated in [1], rings R for which A(R) is a Gödel algebra, also called Gödel rings are investigated in [2] and very recently rings R for which A(R) is a pseudo MV-algebra, also called Generalized Łukasiewicz rings are investigated in [10]. In the same spirit, the present article aims at initiating the study of the class of commutative rings R for which A(R) is a BL-algebra, herein referred to as BL-rings. At first glance, requiring that the ideals of a commutative ring carry a structure of BL-algebra might seem very stringent, but at the same time this class already contains all the Łukasiewicz rings [1]. In addition, it turns out that this class is properly contained in the class of multiplication rings as treated in [8], the class of arithmetical rings as treated in [9] and contains properly each of the classes of Dedekind domains, discrete valuation rings, Noetherian multiplication rings. In addition, this class is found to be closed under finite direct products, arbitrary direct sums and homomorphic images. Furthermore, a description of subdirectly irreducible BL-rings is obtained, which combined with the well-known Birkhoff representation theorem, provides a representation of general BL-rings. We close this preliminary study of BL-rings with several properties of their ideals. All rings treated are commutative. We recall that a commutative integral residuated lattice can be defined as a nonempty set L with four binary operations ∧, ∨, ⊗, → and two constants 0,1 satisfying: (i) $$\mathbb{L}(L):=(L,\wedge , \vee , 0, 1)$$ is a bounded lattice; (ii) (L, ⊗, 1) is a commutative monoid and (iii) (⊗, →) form an adjunct pair, i.e. x ⊗ y ≤ z iff x ≤ y → z, for all x, y, z ∈ L. An Rℓ-monoid is a residuated lattice L satisfying the divisibility axiom, that is: x ∧ y = (x → y) ⊗ x, for all x, y ∈ L. A BL- algebra is an Rℓ monoid L satisfying the pre-linearity axiom, that is: (x → y) ∨ (y → x) = 1, for all x, y ∈ L An MV-algebra is a BL-algebra L satisfying the double negation law: x** = x, for all x ∈ L, where x* = x → 0. Given any BL-algebra L, MV (L) := {x* : x ∈ L} is an MV-algebra, and is indeed the largest BL-sub algebra of L satisfying the double negation. This MV-algebra is called the MV-center of L. A detailed treatment of the MV-center can be found in [15]. Given any commutative ring R generated by idempotents (that is for every x ∈ R, there exists e = e2 ∈ R such that xe = x), the lattice of ideals of R form a residuated lattice A(R) := ⟨Id(R), ∧, ∨, ⊗, →, {0}, R⟩, where I ∧ J = I ∩ J, I ∨ J = I + J, I ⊗ J := I ⋅ J, I → J := {x ∈ R : xI ⊆ J}. Note that I* is simply the annihilator of I in R. The following notations will be used throughout the paper. Given a commutative ring R, recall that an ideal I of R is called an annihilator ideal (resp. a dense ideal) if I = J* for some ideal J of R (resp. I* = {0}): A(R) denotes the residuated lattice of ideals of R; $$MV(R):=\left \{I^{\ast }: I\in A(R)\right \}$$ denotes the set of annihilator ideals of R; D(R) := {I ∈ A(R) : I* = {0}} denotes the set of dense ideals of R. 2 BL-rings, definitions, examples and first properties In this section, we introduce BL-rings which shall be commutative rings generated by idempotents for which the lattices of ideals are naturally equipped with BL-algebra structures. Some of the main properties of these rings and their connections to other known classes of rings shall be established. First and foremost, we formally state the definition. Definition 2.1 A commutative ring R is called a BL-ring if it satisfies, BLR-1: I ∩ J = I ⋅ (I → J) for all ideals I, J of R, and BLR-2: (I → J) + (J → I) = R for all ideals I, J of R. Note that BLR-1 is equivalent to I ∩ J ⊆ I ⋅ (I → J) since the inclusion I ⋅ (I → J) ⊆ I ∩ J holds in any ring. In addition, BLR-2 is easily seen to be equivalent to each of the following conditions: BLR-2.1: (I ∩ J) → K = (I → K) + (J → K) for all ideals I, J, K of R. BLR-2.2: I → (J + K) = (I → J) + (I → K) for all ideals I, J, K of R. A class of commutative rings that is very close to that of BL-rings is the class of multiplication rings. Recall that a commutative ring R is called a multiplication ring if for every ideals I, J of R such that I ⊆ J, there exists an ideal K such that I = J ⋅ K. A commutative ring R is called almost multiplication ring (AM-ring) if for every ideals I, J of R such that $$I\subsetneq J$$, there exists an ideal K such that I = J ⋅ K. Proposition 2.2 A commutative ring satisfies BLR-1 if and only if it is a multiplication ring. Proof. Suppose that R is a BL-ring and let I, J be ideals of R such that I ⊆ J. Then by BLR-1, I = I ∩ J = I ⋅ (I → J). Take K = I → J. Conversely, suppose that R is a multiplication ring and let I, J be ideals of R. Since I ∩ J ⊆ I, there exists an ideal K of R such that I ∩ J = I ⋅ K. Hence, I ⋅ K ⊆ J and it follows that K ⊆ I → J. Thus, I ∩ J ⊆ I ⋅ (I → J). As observed above, the inclusion I ⋅ (I → J) ⊆ I ∩ J holds in any ring. Therefore, BLR-1 holds as needed. Corollary 2.3 Every BL-ring is generated by idempotents. A commutative ring is a BL-ring if and only if A(R) is a BL-algebra. Proof. BLR-1 implies that R is a multiplication ring, and it is known that every multiplication ring is generated by idempotents [8, Cor. 7]. This is clear from (1) and the axioms BLR-1 and BLR-2. Another class of rings that contains all BL-rings is the class of arithmetical rings. Recall [9] that a commutative ring R is called arithmetical if its lattice of ideals is distributive, i.e. I ∩ (J + K) = I ∩ J + I ∩ K for all ideals I, J, K in R. Now suppose that R is a BL-ring, then by Corollary 2.3 (2) A(R) is a BL-algebra and BL-algebras are known to be distributive lattices (see for e.g. [14, Prop. 1]. Therefore R is an arithmetical ring. In fact every ring satisfying BLR-1 is an arithmetical ring. Example 2.4 Here are some important classes of BL-rings. Discrete valuation rings (dvr). The ideals of a dvr are principal and totally ordered by the inclusion. Clearly, BLR-2 holds in any chain ring. As for BLR-1, let I, J be ideals of a dvr R. If I ⊆ J, then I → J = R and I ∩ J = I ⋅ (I → J). On the other hand, if J ⊆ I, since I is principal, then I = aR for some a ∈ R. Let j ∈ J ⊆ aR, then j = ax. So ax ∈ J and x ∈ I → J. Hence, j ∈ I ⋅ (I → J) and I ∩ J ⊆ I ⋅ (I → J). Noetherian multiplication rings. By Proposition 2.2, every multiplication ring satisfies BLR-1. In addition, if R is a Noetherian multiplication ring, then R is a Noetherian arithmetical ring and by [9, Thm. 3], K → (I + J) = (K → I) + (K → J) for all ideals I, J, K of R. Hence, R satisfies BLR-2.2, which is equivalent to BLR-2 as observed earlier. Whence, R is a BL-ring as claimed. Łukasiewicz rings. Indeed, if R is a Łukasiewicz ring, then A(R) is an MV-algebra [1]. Commutative Gödel rings. Indeed, if R is a commutative Gödel ring, then A(R) is a BL-algebra satisfying I ⋅ J = I ∩ J [2]. Recall that a commutative ring R is called a ZPI-ring if each ideal of R can be represented as a finite product of prime ideals. A special primary ring (SPIR) is a ring in which every ideal is a power of a fixed prime ideal. Remark 2.5 Each of the classes of rings of Example 2.4 is a proper subclass of the class of BL-rings. In fact $$\mathbb{Z}$$ is a Noetherian multiplication ring that is neither a dvr nor a Łukasiewicz ring. In addition, $$\oplus _{n=1}^{\infty }\mathbb{R}$$ is a Łukasiewicz ring that is neither Noetherian nor a dvr. A Noetherian ring is a BL-ring if and only if it is a multiplication ring. In addition, note that Noetherian multiplication rings are ZPI-rings (see for e.g. [11, Exercise 10(b), pp. 224]), and ZPI-rings are direct sums of a finite number of Dedekind domains and SPIR. Therefore, Noetherian BL-rings are direct sums of a finite number of Dedekind domains and SPIRs. The following lemma will be needed when working with BL-rings. Lemma 2.6 Let R be a ring, and I, J, K be ideals of R such that I ⊆ J, K. Then, I ⊆ (I*⋅J)*, J → I, J → K, K → J; (J/I)* = (J → I)/I; (J/I) → (K/I) = (J → K)/I. Proof. These are easily derived from the definitions of the operations involved. Note that if a ring satisfies BLR-2, then since I → J = I → (I ∩ J), it must satisfy the following: BLR-3: I ∩ J = {0} implies I* + J* = R for all ideals I, J of R. Proposition 2.7 A ring R satisfies BLR-2 if and only if every quotient (by an ideal) of R satisfies BLR-3. Proof. Suppose that R satisfies BLR-2, and let I be an ideal of R. Let I ⊆ J, K such that (J/I) ∩ (K/I) = I. Then, J ∩ K = I. Now, (J/I)*+(K/I)* = (J → I)/I+(K → I)/I = (J → (J∩K))/I+(K → (J∩K))/I = ((J → K)+(K → J))/I = R/I. Thus, R/I satisfies BLR-3. Conversely, suppose that every factor of R satisfies BLR-3. Let I, J be ideals of R, then R/(I ∩ J) satisfies BLR-3. Since (I/(I ∩ J)) ∩ (J/(I ∩ J)) = I ∩ J, then (I/(I∩J))* + (J/(I∩J))* = R/(I ∩ J). That is, (I → (I ∩ J))/I + (J → (I ∩ J))/I = R/(I ∩ J) or (I → J)/(I ∩ J) + (J → I)/(I ∩ J) = R/(I ∩ J). Thus, ((I → J) + (J → I))/(I ∩ J) = R/(I ∩ J) and it follows that (I → J) + (J → I) = R. So, R satisfies BLR-2 as needed. Corollary 2.8 Every multiplication ring satisfies BLR-2 if and only if every multiplication ring satisfies BLR-3. The following example, which is a special case of (8, Example 4) is a multiplication ring that does not satisfy BLR-2. Example 2.9 Let F be any field and let R be the subring of $$\prod _{k=1}^{\infty} F$$ generated by $$\oplus _{k=1}^{\infty} F$$ and the constant functions from $$\mathbb{N}\to F$$. Then R is a multiplication ring with identity. Note that R is the subring of $$\prod _{k=1}^{\infty } F$$ of all sequences $$\mathbb{N}\to F$$ that are eventually constant. That is f ∈ R if and only if there exists x ∈ F and n ≥ 1 such that f(k) = x for all k ≥ n. Now, let $$I=\{f\in R: f(2k)=0\; \textrm{for all}\; k\in \mathbb{N}\}$$ and $$J=\{f\in R: f(2k+1)=0\; \textrm{for all}\; k\in \mathbb{N}\}$$. Then $$I, J\subseteq \oplus _{k=1}^{\infty } F$$, I ∩ J = {0}, I* = J, J* = I. Thus, I* + J* = I + J≠R. Therefore, BL-rings form a proper subclass of the class of multiplication rings. Proposition 2.10 Let R be a ring that is generated by idempotents and P be a prime ideal of R. Recall [11, §IX.4] that $$ N(P)=\{x\in R:xs=0\; \; \textrm{for some }\; \; s\in R\setminus P\}$$ Then $$\bigcap\nolimits_{P} N(P)=\{0\}$$. Proof. Let x≠0, then (xR)* ≠ R. Thus, there exists a prime ideal P of R such that (xR)* ⊆ P. We claim that x ∉ N(P). Indeed, if x ∈ N(P), then xs = 0 for some s ∉ P. This would imply that s ∈ (xR)*⊆ P, which is a contradiction. Proposition 2.11 Every unitary BL-ring is isomorphic to a subring of a direct product of dvrs and SPIRs. Proof. Let R be a unitary BL-ring, then R is a unitary multiplication ring. Consider $$\varphi :R\to \prod _{P}R_{P}$$ defined by $$\varphi (x)=(\frac{x}{1})_{P}$$. Then φ is a ring homomorphism and ker$$\varphi =\bigcap _{P} N(P)$$. Thus, φ is an injective ring homomorphism by Proposition 2.10, which implies that R is isomorphic to φ(R), a subring of $$\prod _{P}R_{P}$$ . On the other hand, R is an AM-ring [5, Lemma 2.4]. Therefore, by [11, Thm. 9.23, Prop. 9.25, Prop. 9.26], each RP is either a dvr or an SPIR. Proposition 2.12 BL-rings are closed under each of the following operations: (1) finite direct products, (2) arbitrary direct sums and (3) homomorphic images. Proof. (1) Let $$R=\prod _{k=1}^{n}R_{k}$$, where each Rk is a BL-ring. Using the fact that each Rk is generated by idempotents, one gets that any ideal I of R is of the form $$I=\prod _{k=1}^{n}I_{k}$$, where Ik is an ideal of Rk for all k. On the other hand, if $$I=\prod _{k=1}^{n}I_{k}$$ and $$J=\prod _{k=1}^{n}J_{k}$$, one can easily verify the following identities: $$I\cdot J=\prod _{k=1}^{n}I_{k}\cdot J_{k}$$, $$I\to J=\prod _{k=1}^{n}I_{k}\to J_{k}$$, $$I\cap J=\prod _{k=1}^{n}I_{k}\cap J_{k}$$ and $$I+J=\prod _{k=1}^{n}I_{k}+J_{k}$$. From these identities, it becomes clear that R satisfies BLR-1 and BLR-2 since each Rk does. Therefore, R is a BL-ring. (2) This is very similar to the above. Indeed, one proves that ideals of direct sums of BL-rings are direct sums of ideals and the argument goes through as in (1), with each instance of the finite direct product replaced by a direct sum. (3) Let R be a BL-ring and I be an ideal of R. We shall show that R/I is a BL-ring. Recall that ideals of R/I are of the form J/I, where J is an ideal of R with I ⊆ J. Let J, K be ideals of R containing I. Using the properties stated in Proposition 2.2, we have (J/I)∩(K/I) = (J∩K)/I = (J⋅(J → K))/I = (J/I)⋅(J → K)/I = (J/I)⋅((J/I) → (K/I)). Thus, R/I satisfies BLR-1. The verification of BLR-2 is similar. Therefore, R/I is a BL-ring as needed. The following example shows that the class of BL-rings is not closed under arbitrary products. Example 2.13 Consider the ring $$R=\prod _{i=1}^{\infty }\mathbb{Z}_{4}$$. It is clear that $$\mathbb{Z}_{4}$$ is a BL-ring. However, since R is not a multiplication ring [8, Example 3], then by Proposition 2.2, it does not satisfy BLR-1. Therefore, R is not a BL-ring. 3 Connection with Baer rings and Von Neumann rings Recall that a Baer ring is a ring in which every annihilator ideal is generated by an idempotent, i.e. for every ideal I of R, there exists an idempotent e ∈ R such that I* = eR. Furthermore, in every Baer ring R, a* is the unique idempotent element in R such that (aR)* = a*R. An ideal I of a Baer ring is called a Baer-ideal if for every a, b ∈ R such that a − b ∈ I, then a*− b*∈ I. A reduced ring is a ring in which 0 is the only nilpotent element, that is, the only element x ∈ R for which there exists an integer n ≥ 1 such that xn = 0. Proposition 3.1 A reduced ring with identity satisfies BLR-3 if and only if it is a Baer ring. Proof. Suppose R is a reduced ring with identity that satisfies BLR-3. Let I be an ideal of R, then since R is reduced, I ∩ I* = {0}. It follows from BLR-3 that I* + I** = R. Hence, 1 = a + b for some a ∈ I* and b ∈ I**. Thus, a = a.1 = a(a + b) = a2 + ab = a2 and a is idempotent. Now, for every x ∈ I*, x = x.1 = x(a + b) = xa + xb = xa. Therefore, I* = aR and R is a Baer ring. Conversely, suppose that R is a Baer ring and let I, J be ideals such that I ∩ J = {0}. Then I ⊆ J* = eR, for some idempotent e ∈ R. Note that since I ⊆ eR and e is idempotent, then 1 − e ∈ I*. Thus, 1 = (1 − e) + e ∈ I* + I** and I* + I** = R. Corollary 3.2 For every Baer ring R and every Baer-ideals I, J of R, (I → J) + (J → I) = R. Proof. Let R be a Baer ring and I, J be Baer-ideals of R, then I ∩ J is a Baer-ideal. Hence, R/(I ∩ J) is a Baer ring and by Proposition 3.1, satisfies BLR-3. It follows as in the proof of Proposition 2.7, that (I → J) + (J → I) = R. Proposition 3.3 Every quotient (by an ideal) of a multiplication ring is a multiplication ring. Every quotient (by an Baer-ideal) of a Baer ring is a multiplication Baer ring. Proof. Let R be a multiplication ring and I an ideal of R. Let J/I ⊆ K/I be ideals of R/I, then J ⊆ K. Thus, as R is a multiplication ring, there exists an ideal T of R such that J = K ⋅ T. Hence, J/I = (K ⋅ T)/I = K/I ⋅ T/I. Therefore, R/I is a multiplication ring. This follows clearly from [13, Lemma 3]. Recall that a commutative unitary ring is a Von Neumann ring (VNR) if and only if RP is a field for all prime ideals of R. Proposition 3.4 Every VNR is a multiplication ring. Proof. Note that by Proposition 2.2, we simply have to prove BLR-1, which we shall do locally. Let R be a VNR, and let I, J be ideals of R and P a prime ideal of R. We need to show that IP ∩ JP = IP ⋅ (I→J)P. Since RP is a field, then IP = {0} or RP and JP = {0} or RP. The equation is obvious when IP = {0}. We consider the remaining two cases. Case 1: Suppose IP = JP = RP, then J ∩ (R ∖ P)≠∅. But, J ⊆ I → J, so J ∩ (R ∖ P) ⊆ (I → J) ∩ (R ∖ P). Hence (I → J) ∩ (R ∖ P)≠∅ and (I→J)P = RP. Thus, IP ⋅ (I→J)P = RP ⋅ (I→J)P = (I→J)P = RP = IP ∩ JP. Case 2: Suppose IP = RP and JP = {0}, then I ∩ (R ∖ P)≠∅. We need to show that (I → J)P = 0. Since I ∩ (R ∖ P)≠∅, there exists s ∈ I and s ∉ P. Now, let x ∈ (I → J) and t ∉ P. Then, xs ∈ J and since JP = {0}, it follows that xs/t = 0/t. Thus, xst′ = 0 for some t′ ∉ P, which implies that x/t = 0/t. Whence, (I → J)P = {0} as needed. Corollary 3.5 Let R be a unitary VNR. Then R is a BL-ring if and only if it satisfies $$ I_{P}=J_{P}=\{0\}\; \; \textrm{implies}\; \; (I\to J)_{P}=R_{P}$$ for all ideals I, J of R and all prime ideal P of R. Proof. This clearly follows from Proposition 3.4 and its proof. 4 Representation and further properties of BL-rings It is standard in universal algebra to understand algebras of a given variety by focusing on the subdirectly irreducible algebras and appealing to the Birkhoff’s representation theorem. This justifies the need to start our analysis with subdirectly irreducible BL-rings. It is known that for every subdirectly irreducible commutative ring R with minimal ideal M, either R is a field or M2 = {0} (see for e.g. [7]). Proposition 4.1 Let R be a subdirectly irreducible BL-ring with minimal ideal M. Then, M is an annihilator ideal, and The annihilator ideals of R are linearly ordered and finite in number. Proof. (1) Since M is the minimal ideal of R, then M ⊆ M*. But since M≠0, then $$M^{\ast } \neq R$$. Therefore, by the maximality of M, we obtain that M = M* and M is an annihilator ideal. (2) Note that A(R) is a BL-algebra and MV (R) is an MV-algebra, more precisely the MV-center of A(R). Moreover, $$(\bigvee I)^{\ast }=\bigcap I^{\ast }$$, which implies that every subset of MV (R) has an infimum. It follows from this that every subset of MV (R) also has a supremum since $$\bigwedge S=(\bigvee S^{\ast })^{\ast }$$ [6, Lemma 6.6.3]. Thus, MV (R) is a complete MV-algebra. In addition, for every nonzero ideal I of R, M ⊆ I, so I*⊆ M*. Therefore, for every proper ideal J in MV (R), we have J ⊆ M*. This means that MV (R) ∖{R} has a maximum element, namely M* (note that M*≠R since M≠0). To see that (MV (R), ⊆) is a chain, let X, Y ∈ MV (R) such that $$X\nsubseteq Y$$ and $$Y\nsubseteq X$$, then X → Y, Y → X≠R and X → Y, Y → X ⊆ M*. Hence, by the pre-linearity axiom, R = (X → Y ) ∨ (Y → X) ⊆ M*. Hence, M* = R, which is a contradiction. Therefore, (MV (R), ⊆) is an MV-chain as claimed. But the only complete MV-chains are finite Łukasiewicz chains and the standard MV-algebra [0, 1]. The condition MV (R) ∖{R} has a maximum element implies that MV (R) is a finite Łukasiewicz chain, which completes the proof of (2). The following result, which is the analog of [2, Theorem 3.9] sheds light on the structure of a general BL-ring. Theorem 4.2 (A Representation Theorem for BL-rings) Every BL-ring R is a subdirect product of a family {Rr : r ∈ R ∖{0}} of subdirectly irreducible BL-rings satisfying A(Rr) ≅ MV (Rr) ⊕ D(Rr) (the ordinal sum as hoops) for all r ≠ 0. Every ideal of each Rr is either an annihilator ideal or dense. A(R) is a subdirect product of {A(Rr) : r ∈ R ∖{0}}. A(Rr) is a BL-algebra with a unique atom. Proof. Recall that it follows from the most celebrated Birkhoff subdirectly irreducible representation theorem (see for e.g. [3, Thm.8.6]), R is a subdirect product of subdirectly irreducible rings, all of whom are homomorphic images of R. Indeed. for every 0≠r ∈ R, an application of Zorn’s lemma shows that there exists an ideal Ir maximal among ideals that do not contain r. It follows that ∩ Ir = {0}, each factor R/Ir is subdirectly irreducible and R is a subdirect product of the family {R/Ir : r ∈ R ∖{0}}. On the other hand, each R/Ir is a BL-ring by Proposition 2.12(3). Therefore the opening statement of the Proposition is established with Rr = R/Ir. (1) and (2) By Proposition 4.1(2), the MV-center of each A(Rr) is a finite Łukasiewicz chain. It follows from [4, Remark 3.3.2] that for all t, A(Rr) ≅ MV (Rr) ⊕ D(Rr), the ordinal sum of MV (Rr) and D(Rr). The property stated in (2) clearly holds for ideals in MV (Rr) ⊕ D(Rr) for each t. Therefore, every ideal of each Rr is either an annihilator ideal or dense as claimed. This completes the proofs of (1) and (2). (3) To show that A(R) is a subdirect product of {R/Ir : r ∈ R ∖{0}}, consider $$\Theta : A(R)\to \prod _{r\ne 0}A(R/I_{r})$$ defined by Θ(I)(r) = I + Ir mod Ir. To see that Θ is a subdirect embedding of BL-algebras, one needs to check the following: (sd1). For every I, J ∈ A(R), I ∩ J + Ir mod Ir = (I + Ir mod Ir) ∩ (J + Ir mod It) for all r. (sd2). For every I, J ∈ A(R), (I + J) + Ir mod Ir = (I + Ir mod Ir) + (J + Ir mod Ir) for all r. (sd3). For every I, J ∈ A(R), (I → J) + Ir mod Ir = (I + Ir mod Ir) → (J + Ir mod Ir) for all r. (sd4). For every I, J ∈ A(R), IJ + Ir mod Ir = (I + Ir mod Ir)(J + Ir mod Ir) for all r. (sd5). For every I ∈ A(R), if I + Ir = R for all r, then I = R. The conditions (sd1)–(sd4) follow easily from the various definitions. As for (sd5), one can establish this locally at each prime ideal P of R (i.e. IP = RP) using the fact that RP is either a dvr or an SPIR as pointed out in the proof of Proposition 2.11. (4) Finally, the fact that each A(Rr) is a BL-algebra with a unique atom is a consequence of Rr being subdirectly irreducible as announced in the opening paragraph. Corollary 4.3 Let R be a subdirectly irreducible BL-ring. Then For every annihilator ideal I≠R and every dense ideal J, I ⊆ J and J → I = I; For every ideals I, J of R, either I → J or J → I is dense. Proof. As established in the proof of Theorem 4.2, A(R)≅MV (R) ⊕ D(R). The condition stated in (1) follows from the definition of the implication in the ordinal sum of hoops. On the other hand, since A(R) is a BL-algebra, it is known (see for e.g. [4, p. 368]) that the set D(L) of dense elements of any BL-algebra L is an implicative filter and L/D(R) ≅ MV (L). Therefore, A(R)/D(R) ≅ MV (R), and since MV (R) is linearly ordered, we deduce that A(R)/D(R) is linearly ordered. The conclusion now follows from the definitions of order and → on A(R)/D(R). We shall end our study by establishing some (further) properties of BL-rings, most of which also shared by Gödel rings [2, Prop. 4.5]. Proposition 4.4 Let R be a BL-ring. Then (i) Let I be an ideal of R and P a prime ideal of R. Then, I ⊆ P or $$I\rightarrow P=P$$. (ii) Every proper ideal is contained in a prime ideal of R. (iii) If P, Q ⊆ R are prime ideals that are not comparable, then they are comaximal, that is P + Q = R. (iv) If P, Q ⊆ R are distinct minimal prime ideals, then they are comaximal. (v) Let I be a proper ideal of R. Then the prime ideals below I, if any, form a chain. (vi) Suppose R is a local ring, i.e. has a unique maximal ideal. Then the prime ideals of R form a chain and each ideal is a power of a prime ideal. (vii) Suppose P, Q ⊆ R are prime ideals that are not comparable. Then there is a BL-epimorphism from $$A(R)\rightarrow A(R/P\times R/Q)$$. (viii) Suppose the set of minimal primes Min(R) is finite, then R is a finite direct sum of Dedekind domains and SPIRs. (ix) For every comaximal ideals I, J of R, I ∩ J = IJ. (x) For every ideal I of R and every maximal ideal M of R not containing I, I ∩ M = IM. (xi) For every prime ideal P of R, MV (R/P) is the 2-element Boolean algebra. Proof. (i) Let I be an ideal of R and P a prime ideal of R. We have from BL-1 $$I\cap P=I(I\rightarrow P)$$. So $$I(I\rightarrow P)\subseteq P$$. Thus, I ⊆ P or $$I\rightarrow P\subseteq P$$. Clearly $$P\subseteq I\rightarrow P$$ and the result follows. (ii) This holds since every BL-ring is generated by idempotents (Corollary 2.3(1)). In fact, let I be a proper ideal R and let a ∉ I. There is an e ∈ R with e2 = e such that a = ae. It is clear that e ∉ I since a ∉ I. Let P be a maximal ideal with respect to containing I and not e. Suppose now that xy ∈ P where x, y ∉ P. Then e ∈ P + xR and e ∈ P + yR (if e is not in these ideals it would contradict the maximality of P). It follows that e ∈ P + xyR ⊆ P, which is a contradiction. Thus, P is prime. (iii) This follows from the combination of BL-2 and (i), since $$R=(P\rightarrow Q)+(Q\rightarrow P)=P+Q=R$$. (iv) This follows from the fact that distinct minimal primes are not comparable and the use of (iii). (v) Suppose P, Q ⊆ I with P and Q prime ideals. If P and Q are not comparable, we have R = P + Q ⊆ R which contradicts the fact that I is proper. Hence the prime ideals below I, if any, form a chain. (vi) Suppose R is a local ring. We have a unique maximal ideal that contains all the prime ideals of R. Thus, the prime ideals of R form a chain by (v). Now the radical of an ideal I of R is the intersection of all prime ideals containing I. Since the prime ideals form a chain, the intersection of a chain of prime ideals is a prime ideal and it then follows that the radical of I is a prime ideal. Since R is a multiplication ring, it follows that I is a power of a prime ideal [12, Theorem 5]. (vii) Suppose P, Q ⊆ R are prime ideals that are not comparable. We know that R/P and R/Q are BL-algebras (so R/P × R/Q is also a BL-algebra) and P + Q = R. Also, R2 + P = R2 + Q = R. By the Chinese Remainder Theorem the natural map $$R\rightarrow R/P\times R/Q$$ is onto. Thus, the natural map induces naturally a BL-algebra epimorphism from $$A(R)\rightarrow A(R/P\times R/Q)$$. (viii) This holds since R is a multiplication ring and [12, Theorem 11]. (ix) Let I, J be comaximal ideals of R and let x ∈ I ∩ J. Since R is generated by idempotents, there exists an idempotent e such that xe = e. But since I + J = R, there exists i ∈ I and j ∈ J such that e = i + j. Therefore, x = xe = ix + xj ∈ IJ. Thus, I ∩ J ⊆ IJ and the equality follows since the other containment holds in general. (x) This follows from (ix) since I and M are comaximal. (xi) Let P be a prime ideal of R and I be an ideal of R such that $$P\varsubsetneq I$$ (i.e. I/P ≠ {0}). Then it follows from (i) above and Lemma 2.6(2) that $$(I/P)^{\ast }=(I\rightarrow P)/P=P/P=\{0\}$$. Therefore, {0} and R/P are the only annihilator ideals of R/P, or equivalently MV (R/P) ≅ 2. Acknowledgements Thanks are due to Professor Bruce Olberding for the fruitful exchanges about multiplication, Baer and VNRs. Additionally, the authors gratefully acknowledge the referees whose comments improve the quality of the paper. References [1] L. P. Belluce and A. Di Nola . Commutative rings whose ideals form an MV-algebra . Mathematical Logic Quarterly, 55 , 468 -- 486 , 2009 . Google Scholar CrossRef Search ADS [2] L. P. Belluce , A. Di Nola and E. Marchioni . Rings and Gödel algebras . Algebra Universalis, 64, 103 -- 116 , 2010 . Google Scholar CrossRef Search ADS [3] S. Burris and H. P. Sankappanavar . A Course in Universal Algebra . Springer , New York-Heidelberg-Berlin , 1981 . Google Scholar CrossRef Search ADS [4] M. Busaniche and F. Montagna . Hájek’s logic BL and BL-algebras. In Handbook of Mathematical Fuzzy Logic , vol. 1, 2011 . [5] H. S. Butts and R. C. Phillips . Almost multiplication rings . Canadian Journal of Mathematics, 17 , 267 -- 277 , 1965 . Google Scholar CrossRef Search ADS [6] R. Cignoli , I. D’Ottaviano and D. Mundici . Algebraic Foundations of Many-Valued Reasoning , Kluwer Academic , Dordrecht, 2000 . Google Scholar CrossRef Search ADS [7] S. Feigelstock . A note on subdirectly irreducible rings . Bulletin of the Australian Mathematical Society , 29 , 137 -- 141 , 1984 . Google Scholar CrossRef Search ADS [8] M. Griffin . Multiplication rings via their total quotient rings . Canadian Journal of Mathematics, 26, 430 -- 449 , 1974. CrossRef Search ADS [9] C. U. Jensen . Arithmetical rings. Acta Mathematica Academiae Scientiarum Hungaricae, 17, 115 -- 123 , 1996 . [10] A. Kadji , C. Lele and J. B. Nganou . On a noncommutative generalization of Łukasiewicz rings . Journal of Applied Logic , 16, 1 -- 13 , 2016. CrossRef Search ADS [11] M. D. Larsen and P. J. McCarthy . Multiplicative Theory of Ideals . Academic Press , New York and London, 1971 . [12] J. L. Mott . Multiplication rings containing only finitely many minimal prime ideals . Journal of Science of the Hiroshima University Series A-I , 73 -- 83 , 1969. [13] T. P. Speed and M. W. Evans . A note on commutative Baer rings. Journal of the Australian Mathematical Society , 14 , 257 -- 263 , 1972. [14] E. Turunen . BL-algebras of basic fuzzy logic. Mathware & Soft Computing , 6 , 49 -- 61 , 1999. [15] E. Turunen . Hyper-Archimedean BL-algebras are MV-algebras . Mathematical Logic Quarterly, 53 , 170 -- 175 , 2007 . Google Scholar CrossRef Search ADS © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Logic Journal of the IGPL Oxford University Press

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Abstract The main goal of this article is to initiate the study of commutative rings whose lattice of ideals can be equipped with a structure of BL-algebra, referred to henceforth as BL-rings. We obtain a description of such rings in terms of their subdirectly irreducible factors. We also investigate their connections with some prominent classes of commutative rings such as multiplication rings, Baer rings, Dedekind rings and Von Neumann rings. 1 Introduction Given any ring R (commutative or not, with or without unity) generated by idempotents, the semiring of ideals of R under the usual operations form a residuated lattice A(R). In recent articles, several authors have investigated classes of rings for which the residuated lattice A(R) is an algebra of a well-known subvariety of residuated lattices. For instance, rings R for which A(R) is an MV-algebra, also called Łukasiewicz rings are investigated in [1], rings R for which A(R) is a Gödel algebra, also called Gödel rings are investigated in [2] and very recently rings R for which A(R) is a pseudo MV-algebra, also called Generalized Łukasiewicz rings are investigated in [10]. In the same spirit, the present article aims at initiating the study of the class of commutative rings R for which A(R) is a BL-algebra, herein referred to as BL-rings. At first glance, requiring that the ideals of a commutative ring carry a structure of BL-algebra might seem very stringent, but at the same time this class already contains all the Łukasiewicz rings [1]. In addition, it turns out that this class is properly contained in the class of multiplication rings as treated in [8], the class of arithmetical rings as treated in [9] and contains properly each of the classes of Dedekind domains, discrete valuation rings, Noetherian multiplication rings. In addition, this class is found to be closed under finite direct products, arbitrary direct sums and homomorphic images. Furthermore, a description of subdirectly irreducible BL-rings is obtained, which combined with the well-known Birkhoff representation theorem, provides a representation of general BL-rings. We close this preliminary study of BL-rings with several properties of their ideals. All rings treated are commutative. We recall that a commutative integral residuated lattice can be defined as a nonempty set L with four binary operations ∧, ∨, ⊗, → and two constants 0,1 satisfying: (i) $$\mathbb{L}(L):=(L,\wedge , \vee , 0, 1)$$ is a bounded lattice; (ii) (L, ⊗, 1) is a commutative monoid and (iii) (⊗, →) form an adjunct pair, i.e. x ⊗ y ≤ z iff x ≤ y → z, for all x, y, z ∈ L. An Rℓ-monoid is a residuated lattice L satisfying the divisibility axiom, that is: x ∧ y = (x → y) ⊗ x, for all x, y ∈ L. A BL- algebra is an Rℓ monoid L satisfying the pre-linearity axiom, that is: (x → y) ∨ (y → x) = 1, for all x, y ∈ L An MV-algebra is a BL-algebra L satisfying the double negation law: x** = x, for all x ∈ L, where x* = x → 0. Given any BL-algebra L, MV (L) := {x* : x ∈ L} is an MV-algebra, and is indeed the largest BL-sub algebra of L satisfying the double negation. This MV-algebra is called the MV-center of L. A detailed treatment of the MV-center can be found in [15]. Given any commutative ring R generated by idempotents (that is for every x ∈ R, there exists e = e2 ∈ R such that xe = x), the lattice of ideals of R form a residuated lattice A(R) := ⟨Id(R), ∧, ∨, ⊗, →, {0}, R⟩, where I ∧ J = I ∩ J, I ∨ J = I + J, I ⊗ J := I ⋅ J, I → J := {x ∈ R : xI ⊆ J}. Note that I* is simply the annihilator of I in R. The following notations will be used throughout the paper. Given a commutative ring R, recall that an ideal I of R is called an annihilator ideal (resp. a dense ideal) if I = J* for some ideal J of R (resp. I* = {0}): A(R) denotes the residuated lattice of ideals of R; $$MV(R):=\left \{I^{\ast }: I\in A(R)\right \}$$ denotes the set of annihilator ideals of R; D(R) := {I ∈ A(R) : I* = {0}} denotes the set of dense ideals of R. 2 BL-rings, definitions, examples and first properties In this section, we introduce BL-rings which shall be commutative rings generated by idempotents for which the lattices of ideals are naturally equipped with BL-algebra structures. Some of the main properties of these rings and their connections to other known classes of rings shall be established. First and foremost, we formally state the definition. Definition 2.1 A commutative ring R is called a BL-ring if it satisfies, BLR-1: I ∩ J = I ⋅ (I → J) for all ideals I, J of R, and BLR-2: (I → J) + (J → I) = R for all ideals I, J of R. Note that BLR-1 is equivalent to I ∩ J ⊆ I ⋅ (I → J) since the inclusion I ⋅ (I → J) ⊆ I ∩ J holds in any ring. In addition, BLR-2 is easily seen to be equivalent to each of the following conditions: BLR-2.1: (I ∩ J) → K = (I → K) + (J → K) for all ideals I, J, K of R. BLR-2.2: I → (J + K) = (I → J) + (I → K) for all ideals I, J, K of R. A class of commutative rings that is very close to that of BL-rings is the class of multiplication rings. Recall that a commutative ring R is called a multiplication ring if for every ideals I, J of R such that I ⊆ J, there exists an ideal K such that I = J ⋅ K. A commutative ring R is called almost multiplication ring (AM-ring) if for every ideals I, J of R such that $$I\subsetneq J$$, there exists an ideal K such that I = J ⋅ K. Proposition 2.2 A commutative ring satisfies BLR-1 if and only if it is a multiplication ring. Proof. Suppose that R is a BL-ring and let I, J be ideals of R such that I ⊆ J. Then by BLR-1, I = I ∩ J = I ⋅ (I → J). Take K = I → J. Conversely, suppose that R is a multiplication ring and let I, J be ideals of R. Since I ∩ J ⊆ I, there exists an ideal K of R such that I ∩ J = I ⋅ K. Hence, I ⋅ K ⊆ J and it follows that K ⊆ I → J. Thus, I ∩ J ⊆ I ⋅ (I → J). As observed above, the inclusion I ⋅ (I → J) ⊆ I ∩ J holds in any ring. Therefore, BLR-1 holds as needed. Corollary 2.3 Every BL-ring is generated by idempotents. A commutative ring is a BL-ring if and only if A(R) is a BL-algebra. Proof. BLR-1 implies that R is a multiplication ring, and it is known that every multiplication ring is generated by idempotents [8, Cor. 7]. This is clear from (1) and the axioms BLR-1 and BLR-2. Another class of rings that contains all BL-rings is the class of arithmetical rings. Recall [9] that a commutative ring R is called arithmetical if its lattice of ideals is distributive, i.e. I ∩ (J + K) = I ∩ J + I ∩ K for all ideals I, J, K in R. Now suppose that R is a BL-ring, then by Corollary 2.3 (2) A(R) is a BL-algebra and BL-algebras are known to be distributive lattices (see for e.g. [14, Prop. 1]. Therefore R is an arithmetical ring. In fact every ring satisfying BLR-1 is an arithmetical ring. Example 2.4 Here are some important classes of BL-rings. Discrete valuation rings (dvr). The ideals of a dvr are principal and totally ordered by the inclusion. Clearly, BLR-2 holds in any chain ring. As for BLR-1, let I, J be ideals of a dvr R. If I ⊆ J, then I → J = R and I ∩ J = I ⋅ (I → J). On the other hand, if J ⊆ I, since I is principal, then I = aR for some a ∈ R. Let j ∈ J ⊆ aR, then j = ax. So ax ∈ J and x ∈ I → J. Hence, j ∈ I ⋅ (I → J) and I ∩ J ⊆ I ⋅ (I → J). Noetherian multiplication rings. By Proposition 2.2, every multiplication ring satisfies BLR-1. In addition, if R is a Noetherian multiplication ring, then R is a Noetherian arithmetical ring and by [9, Thm. 3], K → (I + J) = (K → I) + (K → J) for all ideals I, J, K of R. Hence, R satisfies BLR-2.2, which is equivalent to BLR-2 as observed earlier. Whence, R is a BL-ring as claimed. Łukasiewicz rings. Indeed, if R is a Łukasiewicz ring, then A(R) is an MV-algebra [1]. Commutative Gödel rings. Indeed, if R is a commutative Gödel ring, then A(R) is a BL-algebra satisfying I ⋅ J = I ∩ J [2]. Recall that a commutative ring R is called a ZPI-ring if each ideal of R can be represented as a finite product of prime ideals. A special primary ring (SPIR) is a ring in which every ideal is a power of a fixed prime ideal. Remark 2.5 Each of the classes of rings of Example 2.4 is a proper subclass of the class of BL-rings. In fact $$\mathbb{Z}$$ is a Noetherian multiplication ring that is neither a dvr nor a Łukasiewicz ring. In addition, $$\oplus _{n=1}^{\infty }\mathbb{R}$$ is a Łukasiewicz ring that is neither Noetherian nor a dvr. A Noetherian ring is a BL-ring if and only if it is a multiplication ring. In addition, note that Noetherian multiplication rings are ZPI-rings (see for e.g. [11, Exercise 10(b), pp. 224]), and ZPI-rings are direct sums of a finite number of Dedekind domains and SPIR. Therefore, Noetherian BL-rings are direct sums of a finite number of Dedekind domains and SPIRs. The following lemma will be needed when working with BL-rings. Lemma 2.6 Let R be a ring, and I, J, K be ideals of R such that I ⊆ J, K. Then, I ⊆ (I*⋅J)*, J → I, J → K, K → J; (J/I)* = (J → I)/I; (J/I) → (K/I) = (J → K)/I. Proof. These are easily derived from the definitions of the operations involved. Note that if a ring satisfies BLR-2, then since I → J = I → (I ∩ J), it must satisfy the following: BLR-3: I ∩ J = {0} implies I* + J* = R for all ideals I, J of R. Proposition 2.7 A ring R satisfies BLR-2 if and only if every quotient (by an ideal) of R satisfies BLR-3. Proof. Suppose that R satisfies BLR-2, and let I be an ideal of R. Let I ⊆ J, K such that (J/I) ∩ (K/I) = I. Then, J ∩ K = I. Now, (J/I)*+(K/I)* = (J → I)/I+(K → I)/I = (J → (J∩K))/I+(K → (J∩K))/I = ((J → K)+(K → J))/I = R/I. Thus, R/I satisfies BLR-3. Conversely, suppose that every factor of R satisfies BLR-3. Let I, J be ideals of R, then R/(I ∩ J) satisfies BLR-3. Since (I/(I ∩ J)) ∩ (J/(I ∩ J)) = I ∩ J, then (I/(I∩J))* + (J/(I∩J))* = R/(I ∩ J). That is, (I → (I ∩ J))/I + (J → (I ∩ J))/I = R/(I ∩ J) or (I → J)/(I ∩ J) + (J → I)/(I ∩ J) = R/(I ∩ J). Thus, ((I → J) + (J → I))/(I ∩ J) = R/(I ∩ J) and it follows that (I → J) + (J → I) = R. So, R satisfies BLR-2 as needed. Corollary 2.8 Every multiplication ring satisfies BLR-2 if and only if every multiplication ring satisfies BLR-3. The following example, which is a special case of (8, Example 4) is a multiplication ring that does not satisfy BLR-2. Example 2.9 Let F be any field and let R be the subring of $$\prod _{k=1}^{\infty} F$$ generated by $$\oplus _{k=1}^{\infty} F$$ and the constant functions from $$\mathbb{N}\to F$$. Then R is a multiplication ring with identity. Note that R is the subring of $$\prod _{k=1}^{\infty } F$$ of all sequences $$\mathbb{N}\to F$$ that are eventually constant. That is f ∈ R if and only if there exists x ∈ F and n ≥ 1 such that f(k) = x for all k ≥ n. Now, let $$I=\{f\in R: f(2k)=0\; \textrm{for all}\; k\in \mathbb{N}\}$$ and $$J=\{f\in R: f(2k+1)=0\; \textrm{for all}\; k\in \mathbb{N}\}$$. Then $$I, J\subseteq \oplus _{k=1}^{\infty } F$$, I ∩ J = {0}, I* = J, J* = I. Thus, I* + J* = I + J≠R. Therefore, BL-rings form a proper subclass of the class of multiplication rings. Proposition 2.10 Let R be a ring that is generated by idempotents and P be a prime ideal of R. Recall [11, §IX.4] that $$ N(P)=\{x\in R:xs=0\; \; \textrm{for some }\; \; s\in R\setminus P\}$$ Then $$\bigcap\nolimits_{P} N(P)=\{0\}$$. Proof. Let x≠0, then (xR)* ≠ R. Thus, there exists a prime ideal P of R such that (xR)* ⊆ P. We claim that x ∉ N(P). Indeed, if x ∈ N(P), then xs = 0 for some s ∉ P. This would imply that s ∈ (xR)*⊆ P, which is a contradiction. Proposition 2.11 Every unitary BL-ring is isomorphic to a subring of a direct product of dvrs and SPIRs. Proof. Let R be a unitary BL-ring, then R is a unitary multiplication ring. Consider $$\varphi :R\to \prod _{P}R_{P}$$ defined by $$\varphi (x)=(\frac{x}{1})_{P}$$. Then φ is a ring homomorphism and ker$$\varphi =\bigcap _{P} N(P)$$. Thus, φ is an injective ring homomorphism by Proposition 2.10, which implies that R is isomorphic to φ(R), a subring of $$\prod _{P}R_{P}$$ . On the other hand, R is an AM-ring [5, Lemma 2.4]. Therefore, by [11, Thm. 9.23, Prop. 9.25, Prop. 9.26], each RP is either a dvr or an SPIR. Proposition 2.12 BL-rings are closed under each of the following operations: (1) finite direct products, (2) arbitrary direct sums and (3) homomorphic images. Proof. (1) Let $$R=\prod _{k=1}^{n}R_{k}$$, where each Rk is a BL-ring. Using the fact that each Rk is generated by idempotents, one gets that any ideal I of R is of the form $$I=\prod _{k=1}^{n}I_{k}$$, where Ik is an ideal of Rk for all k. On the other hand, if $$I=\prod _{k=1}^{n}I_{k}$$ and $$J=\prod _{k=1}^{n}J_{k}$$, one can easily verify the following identities: $$I\cdot J=\prod _{k=1}^{n}I_{k}\cdot J_{k}$$, $$I\to J=\prod _{k=1}^{n}I_{k}\to J_{k}$$, $$I\cap J=\prod _{k=1}^{n}I_{k}\cap J_{k}$$ and $$I+J=\prod _{k=1}^{n}I_{k}+J_{k}$$. From these identities, it becomes clear that R satisfies BLR-1 and BLR-2 since each Rk does. Therefore, R is a BL-ring. (2) This is very similar to the above. Indeed, one proves that ideals of direct sums of BL-rings are direct sums of ideals and the argument goes through as in (1), with each instance of the finite direct product replaced by a direct sum. (3) Let R be a BL-ring and I be an ideal of R. We shall show that R/I is a BL-ring. Recall that ideals of R/I are of the form J/I, where J is an ideal of R with I ⊆ J. Let J, K be ideals of R containing I. Using the properties stated in Proposition 2.2, we have (J/I)∩(K/I) = (J∩K)/I = (J⋅(J → K))/I = (J/I)⋅(J → K)/I = (J/I)⋅((J/I) → (K/I)). Thus, R/I satisfies BLR-1. The verification of BLR-2 is similar. Therefore, R/I is a BL-ring as needed. The following example shows that the class of BL-rings is not closed under arbitrary products. Example 2.13 Consider the ring $$R=\prod _{i=1}^{\infty }\mathbb{Z}_{4}$$. It is clear that $$\mathbb{Z}_{4}$$ is a BL-ring. However, since R is not a multiplication ring [8, Example 3], then by Proposition 2.2, it does not satisfy BLR-1. Therefore, R is not a BL-ring. 3 Connection with Baer rings and Von Neumann rings Recall that a Baer ring is a ring in which every annihilator ideal is generated by an idempotent, i.e. for every ideal I of R, there exists an idempotent e ∈ R such that I* = eR. Furthermore, in every Baer ring R, a* is the unique idempotent element in R such that (aR)* = a*R. An ideal I of a Baer ring is called a Baer-ideal if for every a, b ∈ R such that a − b ∈ I, then a*− b*∈ I. A reduced ring is a ring in which 0 is the only nilpotent element, that is, the only element x ∈ R for which there exists an integer n ≥ 1 such that xn = 0. Proposition 3.1 A reduced ring with identity satisfies BLR-3 if and only if it is a Baer ring. Proof. Suppose R is a reduced ring with identity that satisfies BLR-3. Let I be an ideal of R, then since R is reduced, I ∩ I* = {0}. It follows from BLR-3 that I* + I** = R. Hence, 1 = a + b for some a ∈ I* and b ∈ I**. Thus, a = a.1 = a(a + b) = a2 + ab = a2 and a is idempotent. Now, for every x ∈ I*, x = x.1 = x(a + b) = xa + xb = xa. Therefore, I* = aR and R is a Baer ring. Conversely, suppose that R is a Baer ring and let I, J be ideals such that I ∩ J = {0}. Then I ⊆ J* = eR, for some idempotent e ∈ R. Note that since I ⊆ eR and e is idempotent, then 1 − e ∈ I*. Thus, 1 = (1 − e) + e ∈ I* + I** and I* + I** = R. Corollary 3.2 For every Baer ring R and every Baer-ideals I, J of R, (I → J) + (J → I) = R. Proof. Let R be a Baer ring and I, J be Baer-ideals of R, then I ∩ J is a Baer-ideal. Hence, R/(I ∩ J) is a Baer ring and by Proposition 3.1, satisfies BLR-3. It follows as in the proof of Proposition 2.7, that (I → J) + (J → I) = R. Proposition 3.3 Every quotient (by an ideal) of a multiplication ring is a multiplication ring. Every quotient (by an Baer-ideal) of a Baer ring is a multiplication Baer ring. Proof. Let R be a multiplication ring and I an ideal of R. Let J/I ⊆ K/I be ideals of R/I, then J ⊆ K. Thus, as R is a multiplication ring, there exists an ideal T of R such that J = K ⋅ T. Hence, J/I = (K ⋅ T)/I = K/I ⋅ T/I. Therefore, R/I is a multiplication ring. This follows clearly from [13, Lemma 3]. Recall that a commutative unitary ring is a Von Neumann ring (VNR) if and only if RP is a field for all prime ideals of R. Proposition 3.4 Every VNR is a multiplication ring. Proof. Note that by Proposition 2.2, we simply have to prove BLR-1, which we shall do locally. Let R be a VNR, and let I, J be ideals of R and P a prime ideal of R. We need to show that IP ∩ JP = IP ⋅ (I→J)P. Since RP is a field, then IP = {0} or RP and JP = {0} or RP. The equation is obvious when IP = {0}. We consider the remaining two cases. Case 1: Suppose IP = JP = RP, then J ∩ (R ∖ P)≠∅. But, J ⊆ I → J, so J ∩ (R ∖ P) ⊆ (I → J) ∩ (R ∖ P). Hence (I → J) ∩ (R ∖ P)≠∅ and (I→J)P = RP. Thus, IP ⋅ (I→J)P = RP ⋅ (I→J)P = (I→J)P = RP = IP ∩ JP. Case 2: Suppose IP = RP and JP = {0}, then I ∩ (R ∖ P)≠∅. We need to show that (I → J)P = 0. Since I ∩ (R ∖ P)≠∅, there exists s ∈ I and s ∉ P. Now, let x ∈ (I → J) and t ∉ P. Then, xs ∈ J and since JP = {0}, it follows that xs/t = 0/t. Thus, xst′ = 0 for some t′ ∉ P, which implies that x/t = 0/t. Whence, (I → J)P = {0} as needed. Corollary 3.5 Let R be a unitary VNR. Then R is a BL-ring if and only if it satisfies $$ I_{P}=J_{P}=\{0\}\; \; \textrm{implies}\; \; (I\to J)_{P}=R_{P}$$ for all ideals I, J of R and all prime ideal P of R. Proof. This clearly follows from Proposition 3.4 and its proof. 4 Representation and further properties of BL-rings It is standard in universal algebra to understand algebras of a given variety by focusing on the subdirectly irreducible algebras and appealing to the Birkhoff’s representation theorem. This justifies the need to start our analysis with subdirectly irreducible BL-rings. It is known that for every subdirectly irreducible commutative ring R with minimal ideal M, either R is a field or M2 = {0} (see for e.g. [7]). Proposition 4.1 Let R be a subdirectly irreducible BL-ring with minimal ideal M. Then, M is an annihilator ideal, and The annihilator ideals of R are linearly ordered and finite in number. Proof. (1) Since M is the minimal ideal of R, then M ⊆ M*. But since M≠0, then $$M^{\ast } \neq R$$. Therefore, by the maximality of M, we obtain that M = M* and M is an annihilator ideal. (2) Note that A(R) is a BL-algebra and MV (R) is an MV-algebra, more precisely the MV-center of A(R). Moreover, $$(\bigvee I)^{\ast }=\bigcap I^{\ast }$$, which implies that every subset of MV (R) has an infimum. It follows from this that every subset of MV (R) also has a supremum since $$\bigwedge S=(\bigvee S^{\ast })^{\ast }$$ [6, Lemma 6.6.3]. Thus, MV (R) is a complete MV-algebra. In addition, for every nonzero ideal I of R, M ⊆ I, so I*⊆ M*. Therefore, for every proper ideal J in MV (R), we have J ⊆ M*. This means that MV (R) ∖{R} has a maximum element, namely M* (note that M*≠R since M≠0). To see that (MV (R), ⊆) is a chain, let X, Y ∈ MV (R) such that $$X\nsubseteq Y$$ and $$Y\nsubseteq X$$, then X → Y, Y → X≠R and X → Y, Y → X ⊆ M*. Hence, by the pre-linearity axiom, R = (X → Y ) ∨ (Y → X) ⊆ M*. Hence, M* = R, which is a contradiction. Therefore, (MV (R), ⊆) is an MV-chain as claimed. But the only complete MV-chains are finite Łukasiewicz chains and the standard MV-algebra [0, 1]. The condition MV (R) ∖{R} has a maximum element implies that MV (R) is a finite Łukasiewicz chain, which completes the proof of (2). The following result, which is the analog of [2, Theorem 3.9] sheds light on the structure of a general BL-ring. Theorem 4.2 (A Representation Theorem for BL-rings) Every BL-ring R is a subdirect product of a family {Rr : r ∈ R ∖{0}} of subdirectly irreducible BL-rings satisfying A(Rr) ≅ MV (Rr) ⊕ D(Rr) (the ordinal sum as hoops) for all r ≠ 0. Every ideal of each Rr is either an annihilator ideal or dense. A(R) is a subdirect product of {A(Rr) : r ∈ R ∖{0}}. A(Rr) is a BL-algebra with a unique atom. Proof. Recall that it follows from the most celebrated Birkhoff subdirectly irreducible representation theorem (see for e.g. [3, Thm.8.6]), R is a subdirect product of subdirectly irreducible rings, all of whom are homomorphic images of R. Indeed. for every 0≠r ∈ R, an application of Zorn’s lemma shows that there exists an ideal Ir maximal among ideals that do not contain r. It follows that ∩ Ir = {0}, each factor R/Ir is subdirectly irreducible and R is a subdirect product of the family {R/Ir : r ∈ R ∖{0}}. On the other hand, each R/Ir is a BL-ring by Proposition 2.12(3). Therefore the opening statement of the Proposition is established with Rr = R/Ir. (1) and (2) By Proposition 4.1(2), the MV-center of each A(Rr) is a finite Łukasiewicz chain. It follows from [4, Remark 3.3.2] that for all t, A(Rr) ≅ MV (Rr) ⊕ D(Rr), the ordinal sum of MV (Rr) and D(Rr). The property stated in (2) clearly holds for ideals in MV (Rr) ⊕ D(Rr) for each t. Therefore, every ideal of each Rr is either an annihilator ideal or dense as claimed. This completes the proofs of (1) and (2). (3) To show that A(R) is a subdirect product of {R/Ir : r ∈ R ∖{0}}, consider $$\Theta : A(R)\to \prod _{r\ne 0}A(R/I_{r})$$ defined by Θ(I)(r) = I + Ir mod Ir. To see that Θ is a subdirect embedding of BL-algebras, one needs to check the following: (sd1). For every I, J ∈ A(R), I ∩ J + Ir mod Ir = (I + Ir mod Ir) ∩ (J + Ir mod It) for all r. (sd2). For every I, J ∈ A(R), (I + J) + Ir mod Ir = (I + Ir mod Ir) + (J + Ir mod Ir) for all r. (sd3). For every I, J ∈ A(R), (I → J) + Ir mod Ir = (I + Ir mod Ir) → (J + Ir mod Ir) for all r. (sd4). For every I, J ∈ A(R), IJ + Ir mod Ir = (I + Ir mod Ir)(J + Ir mod Ir) for all r. (sd5). For every I ∈ A(R), if I + Ir = R for all r, then I = R. The conditions (sd1)–(sd4) follow easily from the various definitions. As for (sd5), one can establish this locally at each prime ideal P of R (i.e. IP = RP) using the fact that RP is either a dvr or an SPIR as pointed out in the proof of Proposition 2.11. (4) Finally, the fact that each A(Rr) is a BL-algebra with a unique atom is a consequence of Rr being subdirectly irreducible as announced in the opening paragraph. Corollary 4.3 Let R be a subdirectly irreducible BL-ring. Then For every annihilator ideal I≠R and every dense ideal J, I ⊆ J and J → I = I; For every ideals I, J of R, either I → J or J → I is dense. Proof. As established in the proof of Theorem 4.2, A(R)≅MV (R) ⊕ D(R). The condition stated in (1) follows from the definition of the implication in the ordinal sum of hoops. On the other hand, since A(R) is a BL-algebra, it is known (see for e.g. [4, p. 368]) that the set D(L) of dense elements of any BL-algebra L is an implicative filter and L/D(R) ≅ MV (L). Therefore, A(R)/D(R) ≅ MV (R), and since MV (R) is linearly ordered, we deduce that A(R)/D(R) is linearly ordered. The conclusion now follows from the definitions of order and → on A(R)/D(R). We shall end our study by establishing some (further) properties of BL-rings, most of which also shared by Gödel rings [2, Prop. 4.5]. Proposition 4.4 Let R be a BL-ring. Then (i) Let I be an ideal of R and P a prime ideal of R. Then, I ⊆ P or $$I\rightarrow P=P$$. (ii) Every proper ideal is contained in a prime ideal of R. (iii) If P, Q ⊆ R are prime ideals that are not comparable, then they are comaximal, that is P + Q = R. (iv) If P, Q ⊆ R are distinct minimal prime ideals, then they are comaximal. (v) Let I be a proper ideal of R. Then the prime ideals below I, if any, form a chain. (vi) Suppose R is a local ring, i.e. has a unique maximal ideal. Then the prime ideals of R form a chain and each ideal is a power of a prime ideal. (vii) Suppose P, Q ⊆ R are prime ideals that are not comparable. Then there is a BL-epimorphism from $$A(R)\rightarrow A(R/P\times R/Q)$$. (viii) Suppose the set of minimal primes Min(R) is finite, then R is a finite direct sum of Dedekind domains and SPIRs. (ix) For every comaximal ideals I, J of R, I ∩ J = IJ. (x) For every ideal I of R and every maximal ideal M of R not containing I, I ∩ M = IM. (xi) For every prime ideal P of R, MV (R/P) is the 2-element Boolean algebra. Proof. (i) Let I be an ideal of R and P a prime ideal of R. We have from BL-1 $$I\cap P=I(I\rightarrow P)$$. So $$I(I\rightarrow P)\subseteq P$$. Thus, I ⊆ P or $$I\rightarrow P\subseteq P$$. Clearly $$P\subseteq I\rightarrow P$$ and the result follows. (ii) This holds since every BL-ring is generated by idempotents (Corollary 2.3(1)). In fact, let I be a proper ideal R and let a ∉ I. There is an e ∈ R with e2 = e such that a = ae. It is clear that e ∉ I since a ∉ I. Let P be a maximal ideal with respect to containing I and not e. Suppose now that xy ∈ P where x, y ∉ P. Then e ∈ P + xR and e ∈ P + yR (if e is not in these ideals it would contradict the maximality of P). It follows that e ∈ P + xyR ⊆ P, which is a contradiction. Thus, P is prime. (iii) This follows from the combination of BL-2 and (i), since $$R=(P\rightarrow Q)+(Q\rightarrow P)=P+Q=R$$. (iv) This follows from the fact that distinct minimal primes are not comparable and the use of (iii). (v) Suppose P, Q ⊆ I with P and Q prime ideals. If P and Q are not comparable, we have R = P + Q ⊆ R which contradicts the fact that I is proper. Hence the prime ideals below I, if any, form a chain. (vi) Suppose R is a local ring. We have a unique maximal ideal that contains all the prime ideals of R. Thus, the prime ideals of R form a chain by (v). Now the radical of an ideal I of R is the intersection of all prime ideals containing I. Since the prime ideals form a chain, the intersection of a chain of prime ideals is a prime ideal and it then follows that the radical of I is a prime ideal. Since R is a multiplication ring, it follows that I is a power of a prime ideal [12, Theorem 5]. (vii) Suppose P, Q ⊆ R are prime ideals that are not comparable. We know that R/P and R/Q are BL-algebras (so R/P × R/Q is also a BL-algebra) and P + Q = R. Also, R2 + P = R2 + Q = R. By the Chinese Remainder Theorem the natural map $$R\rightarrow R/P\times R/Q$$ is onto. Thus, the natural map induces naturally a BL-algebra epimorphism from $$A(R)\rightarrow A(R/P\times R/Q)$$. (viii) This holds since R is a multiplication ring and [12, Theorem 11]. (ix) Let I, J be comaximal ideals of R and let x ∈ I ∩ J. Since R is generated by idempotents, there exists an idempotent e such that xe = e. But since I + J = R, there exists i ∈ I and j ∈ J such that e = i + j. Therefore, x = xe = ix + xj ∈ IJ. Thus, I ∩ J ⊆ IJ and the equality follows since the other containment holds in general. (x) This follows from (ix) since I and M are comaximal. (xi) Let P be a prime ideal of R and I be an ideal of R such that $$P\varsubsetneq I$$ (i.e. I/P ≠ {0}). Then it follows from (i) above and Lemma 2.6(2) that $$(I/P)^{\ast }=(I\rightarrow P)/P=P/P=\{0\}$$. Therefore, {0} and R/P are the only annihilator ideals of R/P, or equivalently MV (R/P) ≅ 2. Acknowledgements Thanks are due to Professor Bruce Olberding for the fruitful exchanges about multiplication, Baer and VNRs. Additionally, the authors gratefully acknowledge the referees whose comments improve the quality of the paper. References [1] L. P. Belluce and A. Di Nola . Commutative rings whose ideals form an MV-algebra . Mathematical Logic Quarterly, 55 , 468 -- 486 , 2009 . Google Scholar CrossRef Search ADS [2] L. P. Belluce , A. Di Nola and E. Marchioni . Rings and Gödel algebras . Algebra Universalis, 64, 103 -- 116 , 2010 . Google Scholar CrossRef Search ADS [3] S. Burris and H. P. Sankappanavar . A Course in Universal Algebra . Springer , New York-Heidelberg-Berlin , 1981 . Google Scholar CrossRef Search ADS [4] M. Busaniche and F. Montagna . Hájek’s logic BL and BL-algebras. In Handbook of Mathematical Fuzzy Logic , vol. 1, 2011 . [5] H. S. Butts and R. C. Phillips . Almost multiplication rings . Canadian Journal of Mathematics, 17 , 267 -- 277 , 1965 . Google Scholar CrossRef Search ADS [6] R. Cignoli , I. D’Ottaviano and D. Mundici . Algebraic Foundations of Many-Valued Reasoning , Kluwer Academic , Dordrecht, 2000 . Google Scholar CrossRef Search ADS [7] S. Feigelstock . A note on subdirectly irreducible rings . Bulletin of the Australian Mathematical Society , 29 , 137 -- 141 , 1984 . Google Scholar CrossRef Search ADS [8] M. Griffin . Multiplication rings via their total quotient rings . Canadian Journal of Mathematics, 26, 430 -- 449 , 1974. CrossRef Search ADS [9] C. U. Jensen . Arithmetical rings. Acta Mathematica Academiae Scientiarum Hungaricae, 17, 115 -- 123 , 1996 . [10] A. Kadji , C. Lele and J. B. Nganou . On a noncommutative generalization of Łukasiewicz rings . Journal of Applied Logic , 16, 1 -- 13 , 2016. CrossRef Search ADS [11] M. D. Larsen and P. J. McCarthy . Multiplicative Theory of Ideals . Academic Press , New York and London, 1971 . [12] J. L. Mott . Multiplication rings containing only finitely many minimal prime ideals . Journal of Science of the Hiroshima University Series A-I , 73 -- 83 , 1969. [13] T. P. Speed and M. W. Evans . A note on commutative Baer rings. Journal of the Australian Mathematical Society , 14 , 257 -- 263 , 1972. [14] E. Turunen . BL-algebras of basic fuzzy logic. Mathware & Soft Computing , 6 , 49 -- 61 , 1999. [15] E. Turunen . Hyper-Archimedean BL-algebras are MV-algebras . Mathematical Logic Quarterly, 53 , 170 -- 175 , 2007 . Google Scholar CrossRef Search ADS © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

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Published: Feb 28, 2018

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