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On right chain ordered semigroups
On right chain ordered semigroups
Changphas, Thawhat; Luangchaisri, Panuwat; Mazurek, Ryszard
2017-09-07 00:00:00
Semigroup Forum (2018) 96:523–535 https://doi.org/10.1007/s00233-017-9896-z RESEARCH ARTICLE 1 1 Thawhat Changphas · Panuwat Luangchaisri · Ryszard Mazurek Received: 8 March 2016 / Accepted: 19 August 2017 / Published online: 7 September 2017 © The Author(s) 2017. This article is an open access publication Abstract A right chain ordered semigroup is an ordered semigroup whose right ideals form a chain. In this paper we study the ideal theory of right chain ordered semigroups in terms of prime ideals, completely prime ideals and prime segments, extending to these semigroups results on right chain semigroups proved in Ferrero et al. (J Algebra 292:574–584, 2005). Keywords Right chain ordered semigroup · Prime ideal · Completely prime ideal · Semiprime ideal · Completely semiprime ideal · Prime segment 1 Introduction and preliminaries Problems studied in this paper have their roots in the theory of chain rings. Recall that a ring R with unity is said to be a right (respectively left) chain ring if its right (respectively left) ideals form a chain, i.e., are totally ordered by set inclusion. If R Communicated by Mikhail Volkov. B Ryszard Mazurek r.mazurek@pb.edu.pl Thawhat Changphas thacha@kku.ac.th Panuwat Luangchaisri desparadoskku@hotmail.com Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand Faculty of Computer Science, Bialystok University of Technology, Wiejska 45A, 15–351 Białystok, Poland 123 524 T. Changphas et al. is a right and left chain ring, then R is called a chain ring. These rings are natural generalizations of commutative valuation rings to the noncommutative case and they have been extensively studied in many papers. In 1976, Brungs and Törner proved in [6, Theorem 3.6] that a semi-invariant chain ring is invariant provided it satisﬁes d.c.c. for prime ideals. This result indicated the importance of the structure of the lattice of prime ideals of a chain ring. In [1, Theorem 3.5], Bessenrodt, Brungs and Törner noted that in a right chain ring, a prime ideal which is not completely prime is always pairing with a unique completely prime ideal, and this result drew attention to the structure of the lattice of completely prime ideals of a right chain ring. As noted in [7], essential for the understanding of the ideal theory of right chain rings R is the understanding of the ideals between two neighbouring completely prime ideals P ⊃ P of R; such a pair ( P , P ) is called a prime segment 1 2 1 2 of R.In[7, Theorem 2.2], Brungs and Törner proved that a prime segment of a right chain ring falls in exactly one of three classes: it is either archimedean, or simple, or exceptional. An analogous classiﬁcation for prime segments of Dubrovin valuation rings was obtained by Brungs et al. [2], and for so called semiprime segments of any ring by Törner and the third author in [13]. Further results on prime segments can be found in, e.g., [3–5,12]. A natural generalization of right chain rings are right chain semigroups, i.e., semi- groups with unity whose right ideals form a chain. Examples of right chain semigroups include the cones of left ordered groups and the multiplicative semigroups of right chain rings. In [8] Brungs and Törner extended the ideal theory of right chain rings in terms of prime ideals, completely prime ideals and prime segments to right cones, that is to right chain semigroups with a left cancellation law. In particular, in [8, Theorem 1.14] Brungs and Törner classiﬁed prime segments of a right cone as either archimedean, or simple, or exceptional. An analogous classiﬁcation of prime segments P ⊃ P for right P -comparable semigroups was obtained by Halimi in [11, Theorem 1 2 1 4.8]. In [9], Ferrero, Sant’Ana, and the third author generalized the ideal theory of right cones to right chain semigroups, but in this case it was necessary to add to the three known types of prime segments (archimedean, simple, exceptional) another one, which was named “supplementary”. In this paper we do a next step, namely we introduce right chain ordered semigroups and extend to them the ideal theory of right chain semigroups developed in [9]. Below in this section we explain why the new class of semigroups is a generalization of right chain semigroups. Recall (see, e.g., [10]) that an ordered semigroup (S, ·, ≤) is a semigroup (S, ·) together with a partial order ≤ that is compatible with the semigroup operation, i.e., for any x , y, z ∈ S we have x ≤ y ⇒ xz ≤ yz and zx ≤ zy. For nonempty subsets A, B of S we deﬁne ( A]={s ∈ S : s ≤ a for some a ∈ A} and AB ={ab : a ∈ A, b ∈ B}. 123 On right chain ordered semigroups 525 A nonempty subset I of an ordered semigroup (S, ·, ≤) is called a right (respec- tively, left) ideal of S if it satisﬁes the following conditions: (1) IS ⊆ I (respectively, SI ⊆ I ); (2) I = (I ], that is, for any s ∈ S and a ∈ I , s ≤ a implies s ∈ I . If I is both a left and a right ideal of S, then I is called a two-sided ideal of S,or simply an ideal of S. Deﬁnition 1.1 An ordered semigroup (S, ·, ≤) is called a right chain ordered semi- group if the right ideals of S form a chain, i.e., for any right ideals I, J of S we have I ⊆ J or J ⊆ I . Left chain semigroups are deﬁned analogously, and S is a chain ordered semigroup if it is a right and left chain ordered semigroup. Note that any semigroup (S, ·) is an ordered semigroup with respect to the trivial order ≤ on S (i.e., the order deﬁned by x ≤ y ⇔ x = y). Furthermore, if ≤ is the trivial order on S, then a subset A ⊆ S is a right ideal of the ordered semigroup (S, ·, ≤) if and only if A is a right ideal of the semigroup (S, ·). Hence right chain semigroups are exactly right chain ordered semigroups with respect to the trivial order. Therefore, the notion of a right chain ordered semigroup generalizes the notion of a right chain semigroup. It is also obvious that any general result on right chain ordered semigroups (S, ·, ≤), when applied to the trivial order ≤ on S, gives its counterpart for the right chain semigroup (S, ·). An element e of an ordered semigroup (S, ·, ≤) is called an identity element of S if ex = x = xe for any x ∈ S. An element 0 of S is called a zero element of S if 0x = 0 = 0x for any x ∈ S. In this paper we assume that each ordered semigroup is with identity element e, and with zero element 0, and e = 0. The following example shows that a right chain ordered semigroup need not be a right chain semigroup. Example 1.2 The set T ={a, b} with the multiplication xy = x for any x , y ∈ T is a semigroup (without zero and identity). Let S be the semigroup obtained from T by adjoining zero and unity elements, i.e., S ={0, e, a, b} and the multiplication in S is deﬁned as follows: x if x = e and y = 0, xy = y if x = e or y = 0. Then {0, a} and {0, b} are incomparable right ideals of the semigroup (S, ·) and thus S is not a right chain semigroup. On the other hand, S (with the above multiplication) is an ordered semigroup with respect to the order x ≤ y ⇔ x = y or (x , y) = (a, b), and the only right ideals of the ordered semigroup (S, ·, ≤) are {0}⊂{0, a}⊂{0, a, b}⊂ S. Hence (S, ·, ≤) is a right chain ordered semigroup. 123 526 T. Changphas et al. The paper is organized as follows. In Sect. 2 we study relationships between prime, semiprime, completely prime, and completely semiprime right ideals of an ordered semigroup (S, ·, ≤), and we provide some methods for constructing such right ideals. In Sect. 3 we show that if furthermore (S, ·, ≤) is a right chain ordered semigroup, then for any right ideal of S, being semiprime is equivalent to being prime, and being completely semiprime is the same as being completely prime. Moreover, in Sect. 3 we use powers of an ideal and powers of an element to construct completely prime right ideals. In Sect. 4 we focus on prime segments of right chain ordered semigroups, prov- ing that any prime segment falls into one of four categories: it has to be archimedean, or simple, or exceptional, or supplementary. In the paper the symbol ⊂ denotes proper inclusion of sets. The set of positive integers is denoted by N. 2 Prime, semiprime, completely prime and completely semiprime ideals of ordered semigroups Let (S, ·, ≤) be an ordered semigroup and let A and B be nonempty subsets of S. Recall that AB denotes the set of all products ab, where a ∈ A and b ∈ B.If s ∈ S, then we write sA (respectively, As) instead of {s} A (respectively, A{s}). If n ∈ N, then A denotes the set of all products a a ··· a , where a , a ,..., a ∈ A.The 1 2 n 1 2 n symbol ( A] denotes the set of all elements s ∈ S such that s ≤ a for some a ∈ A. In the following proposition we record some basic properties of these sets. We will freely use this proposition in the paper; its easy proof is left to the reader. Proposition 2.1 Let (S, ·, ≤) be an ordered semigroup. (1) For any nonempty subsets A, B of S, the following hold: (a) A ⊆ ( A] and (( A]] = ( A], (b) If A ⊆ B, then ( A]⊆ (B], (c) (( A](B]] = (( A]B]= ( A(B]] = (AB]. Consequently, ( A](B]⊆ (AB], ( A]B ⊆ (AB], and A(B]⊆ (AB], m n mn (d) (( A ] ]= ( A ] for any m, n ∈ N, (e) If x , y ∈ S and x ≤ y, then (xA]⊆ (yA] and ( Ax]⊆ ( Ay]. (2) If { A } is a family of nonempty subsets of S, then ( A ]= ( A ] k k∈K k k k∈K k∈K and ( A ]⊆ ( A ]. k k k∈K k∈K It is easy to see that if I and J are right (respectively, two-sided) ideals of S, then (IJ ] is a right (respectively, two-sided) ideal of S. Furthermore, directly from Proposition 2.1(2) we obtain the following corollary. Corollary 2.2 If {I } is a family of right (resp. left, two-sided) ideals of an ordered k k∈K semigroup (S, ·, ≤), then I and I are right (resp. left, two-sided) ideals k k k∈K k∈K of S. Let (S, ·, ≤) be an ordered semigroup. A right ideal I of S is said to be – prime if I is proper and for any right ideals A, B of S, AB ⊆ I implies A ⊆ I or B ⊆ I ; 123 On right chain ordered semigroups 527 – completely prime if I is proper and for any elements a, b of S, ab ∈ I implies a ∈ I or b ∈ I ; – semiprime if I is proper and for any right ideal A of S, A ⊆ I implies A ⊆ I ; – completely semiprime if I is proper and for any element a of S, a ∈ I implies a ∈ I . From the above deﬁnitions we obtain immediately the following implication chart for the considered types of right ideals: prime ⇒ semiprime ⇑⇑ (2.1) completely prime ⇒ completely semiprime Below we prove another interrelation between considered types of ideals. Proposition 2.3 Let I be an ideal of an ordered semigroup (S, ·, ≤). Then I is com- pletely prime if and only if I is prime and completely semiprime. Proof Obviously if I is completely prime, then I is prime and completely semiprime. To prove the converse implication, assume that I is prime and completely semiprime. To show that I is completely prime, consider any a, b ∈ S such that ab ∈ I . Then (bSa] = (bSa](bSa]⊆ (bSabSa]⊆ (I]= I, and since I is completely semiprime, we obtain (bSa]⊆ I and thus bSa ⊆ I . Hence (bS](aS]⊆ (bSaS]⊆ (I]= I, (2.2) and since I is prime, it follows from (2.2) that (bS]⊆ I or (aS]⊆ I . Thus b ∈ I or a ∈ I , which shows that I is completely prime. The following concept will be useful in constructing semiprime ideals of an ordered semigroup (S, ·, ≤). For any proper right ideal A of S we deﬁne the Hoehnke ideal of S associated with A to be the set H (S) ={h ∈ S : s ∈ / (shS] for all s ∈ S\ A}. If the order ≤ is trivial and A is an ideal of S, then the Hoehnke ideal coincides with the set H (S) ={h ∈ S : s ∈ / shS f or all s ∈ S\ A}, which was deﬁned and studied in [9] (for information why Hoehnke’s name appears in this context, the interested reader is referred to [9]). Below we extend [9, Proposition 2] to ordered semigroups, showing in particular that indeed H (S) is an ideal of S. Theorem 2.4 Let A be a proper right ideal of an ordered semigroup (S, ·, ≤). Then (1) H (S) is a semiprime ideal of S. (2) For any right ideal I of S, I ⊆ H (S) if and only if s ∈ / (sI ] for all s ∈ S\ A. (3) If A is an ideal of S, then A ⊆ H (S). 123 528 T. Changphas et al. Proof We show ﬁrst that H (S) is an ideal of S. Since 0 ∈ A, (s0S]= (0]⊆ ( A]= A for any s ∈ S. Thus s ∈ / (s0S] for all s ∈ S\ A, which shows that 0 ∈ H (S). Hence the set H (S) is nonempty. To show that H (S) is closed under multiplication (from both A A the left and the right) by elements of S, suppose for a contradiction that t ht ∈ / H (S) 1 2 A for some h ∈ H (S) and t , t ∈ S. Then there exists s ∈ S\ A such that A 1 2 s ∈ (st ht S]. (2.3) 1 2 From (2.3) we get st ∈ (st ht S]S ⊆ (st ht S]⊆ (st hS], and since h ∈ H (S), 1 1 2 1 2 1 A it follows that st ∈ A. Thus (st ht S]⊆ ( A]= A,so(2.3) implies s ∈ A,a 1 1 2 contradiction. Hence SH (S)S ⊆ H (S). To complete the proof that H (S) is an ideal A A A of S, it sufﬁces to show that (H (S)]⊆ H (S). For this, consider any y ∈ (H (S)]. A A A Then y ≤ h for some h ∈ H (S). Since for any s ∈ S\ A we have sy ≤ sh, we get (syS]⊆ (shS].Now h ∈ H (S) implies s ∈ / (shS],soalso s ∈ / (syS], and y ∈ H (S) A A follows. Hence (H (S)]⊆ H (S), as desired. A A Before showing that the ideal H (S) is semiprime, we prove for any right ideal I of S the equivalence stated in (2). We proceed by contraposition. If I H (S), then there exist i ∈ I and s ∈ S\ A such that s ∈ (si S], and s ∈ (sI ] follows, which proves the implication “⇐” in (2). To prove the opposite implication, suppose that s ∈ (sI ] for some s ∈ S\ A. Then for some i ∈ I we have s ≤ si = sie ∈ si S, and thus s ∈ (si S], which implies that i ∈ / H (S). Hence I H (S) and the proof of (2) is A A complete. To establish (1), it remains to show that the ideal H (S) is semiprime. Since e ∈ / H (S), the ideal H (S) is proper. Let I be a right ideal of S such that I ⊆ H (S).If A A A I H (S), then by (2) there exists s ∈ S\ A such that s ∈ (sI ]. Hence sI ⊆ (sI ]I ⊆ 2 2 (sI ],so (sI]⊆ (sI ], and thus s ∈ (sI]⊆ (sI ]⊆ (sH (S)], which is a contradiction by (2). Thus I ⊆ H (S) and therefore H (S) is a semiprime A A ideal of S. The prove of (1) is complete. To prove (3), assume that A is an ideal of S. Then (sA]⊆ ( A]= A for any s ∈ S, and thus s ∈ / (sA] for all s ∈ S\ A. Hence (2) implies A ⊆ H (S), as desired. For any proper right ideal A of an ordered semigroup (S, ·, ≤) we deﬁne the asso- ciated prime right ideal of A to be the set P ( A) ={ p ∈ S : sp ∈ Afor some s ∈ S\ A}. This concept is an analogue of the notion introduced in [9, Deﬁnition 12]. Below we extend [9, Lemma 13(i)] to ordered semigroups. Proposition 2.5 Let A be a proper right ideal of an ordered semigroup (S, ·, ≤). Then P ( A) is a completely prime right ideal of S containing A. Proof By assumption, A is a proper right ideal of S. Hence e ∈ / A, and since for any a ∈ A we have ea = a ∈ A, it follows that A ⊆ P ( A). Now we show that P ( A) is a r r 123 On right chain ordered semigroups 529 right ideal of S. For this, let us consider any s ∈ S and p ∈ P ( A). Since p ∈ P ( A), r r for some x ∈ S\ A we have xp ∈ A, and since A is a right ideal of S, we obtain x ( ps) = (xp)s ∈ A and thus ps ∈ P ( A). Hence P ( A)S ⊆ P ( A). To complete r r r the proof that P ( A) is a right ideal of S, it sufﬁces to show that ( P ( A)]⊆ P ( A). r r r For this, consider any q ∈ ( P ( A)]. Then there exists p ∈ S such that q ≤ p and xp ∈ A for some x ∈ S\ A. Since xq ≤ xp and xp ∈ A,itfollows that xq ∈ ( A]= A and thus q ∈ P ( A), as desired. Since obviously e ∈ / P ( A), it follows that P ( A) is r r r a proper right ideal of S. To show that this right ideal is completely prime, consider any a, b ∈ S with ab ∈ P ( A). Then there exists x ∈ S\ A such that xab ∈ A.If xa ∈ A, then we have a ∈ P ( A). Otherwise xa ∈ S\ A, and since (xa)b = xab ∈ A, b ∈ P ( A) follows. 3 Prime and completely prime ideals of right chain ordered semigroups The following result shows that for any right chain ordered semigroup the horizontal implications on the chart (2.1) are in fact equivalences. The result is a generalization of [9, Lemma 8]. Proposition 3.1 If (S, ·, ≤) is a right chain ordered semigroup, then (1) A right ideal I of S is semiprime if and only if I is prime. (2) An ideal I of S is completely semiprime if and only if I is completely prime. Proof (1) Assume that I is a semiprime right ideal of S.Let A, B be right ideals of S such that AB ⊆ I . Since S is a right chain ordered semigroup, we must have A ⊆ B or B ⊆ A.If A ⊆ B, then A ⊆ AB ⊆ I and A ⊆ I follows. Similarly B ⊆ A implies B ⊆ I . Thus I is prime. The converse statement is obvious. (2) If an ideal I is completely semiprime, then (1) implies that I is prime, and thus I is completely prime by Proposition 2.3. The converse statement is clear. Let A be an ideal of an ordered semigroup (S, ·, ≤). We adopt from [9] the following two useful notions. An ideal I of S is said to be A-nilpotent if I ⊆ A for some n ∈ N. An element t of S is said to be A-nilpotent if t ∈ A for some n ∈ N. The following result extends [9, Proposition 9] to right chain ordered semigroups. Proposition 3.2 Let A be a proper ideal of a right chain ordered semigroup (S, ·, ≤ ). (1) If I is an ideal of S such that I ⊆ H (S) and I is not A-nilpotent, then (I ] n∈N is a completely prime ideal of S. (2) If t ∈ S is such that t ∈ H (S) and t is not A-nilpotent, then (t S] is a n∈N prime right ideal of S. Proof (1) Assume that I is an ideal of S such that I ⊆ H (S) and I is not A- nilpotent. Since H (S) is a proper ideal of S by Theorem 2.4(1), so is I and thus Corollary 2.2 implies that (I ] is a proper ideal of S. By Proposition 3.1(2), n∈N 2 n to complete the proof of (1), it sufﬁces to show that for any a ∈ S, a ∈ (I ] n∈N implies a ∈ (I ]. For a contradiction, assume that there exists a ∈ S such that n∈N 123 530 T. Changphas et al. 2 n n m a ∈ (I ] but a ∈ / (I ]. Then a ∈ / (I ] for some m ∈ N. Since S is a right n∈N n∈N chain ordered semigroup, we must have (I ]⊆ (aS]. Hence 2 2m+1 m m+1 m+1 a ∈ (I ]= (I I ]⊆ ((aS]I ] m+1 m 2 ⊆ (aI ]⊆ (aI I]⊆ (a(aS]I]⊆ (a I ], (3.1) and Theorem 2.4(2) implies a ∈ A. Thus from (3.1) we obtain 2m+1 2m+1 2 I ⊆ (I ]⊆ (a I]⊆ ( A]= A, so I is A-nilpotent. This contradiction completes the proof of (1). (2) Assume t ∈ H (S) and t is not A-nilpotent. Since t ∈ H (S), it follows A A from Theorem 2.4(1) that (t S]⊆ H (S) and thus Corollary 2.2 implies that n∈N (t S] is a proper right ideal of S. By Proposition 3.1(1), to complete the proof of n∈N 2 n (2), it is enough to show for any right ideal J of S that J ⊆ (t S] implies J ⊆ n∈N n 2 n n (t S]. Suppose for a contradiction that J ⊆ (t S] but J (t S]. n∈N n∈N n∈N Then J (t S] for some m ∈ N, and since S is a right chain ordered semigroup, we have (t S]⊆ J . Hence 2m m m 2 2m+1 2m 2m t ∈ (t S](t S]⊆ J ⊆ (t S]= (t tS]⊆ (t (tS]] 2m and thus t ∈ A by Theorem 2.4(2). But this is a contradiction, since t is not A- nilpotent. Example 10 in [9] shows that in Proposition 3.2 the assumptions I ⊆ H (S) in part (1) and t ∈ H (S) in part (2) are both necessary. The following corollary generalizes [9, Corollary 11]. Corollary 3.3 Let I be an ideal of a right chain ordered semigroup (S, ·, ≤) such that n n+1 n (I ] = (I ] for any n ∈ N. Then (I ] is a completely prime ideal of S. n∈N Proof By Corollary 2.2, A = (I ] is an ideal of S. By Proposition 3.2(1), to n∈N prove that the ideal A is completely prime it sufﬁces to show that the ideal A is proper, I ⊆ H (S) and I is not A-nilpotent. If A = S, then for any n ∈ N we have (I ]= S, n n+1 hence (I ]= (I ], and this contradiction shows that the ideal A is proper. Let m 2 s ∈ S\ A. Then s ∈ / (I ] for some m ∈ N.If s ∈ (sI ], then (sI]⊆ ((sI ]I]⊆ (sI ],so 2 m m s ∈ (sI ] and continuing this way we obtain s ∈ (sI ]⊆ (I ], a contradiction. Hence s ∈ / (sI ], and thus I ⊆ H (S) by Theorem 2.4(2). To show that I is not A-nilpotent, k k suppose for a contradiction that I ⊆ A for some k ∈ N. Then (I ]⊆ ( A]= A, and k k+1 k k k+1 thus (I ]⊆ A ⊆ (I ]⊆ (I ], which implies (I ]= (I ], a contradiction. Thus I is not A-nilpotent. Later on we will need the following generalization of [9, Lemma 13(ii)]. Proposition 3.4 Let A be a prime right ideal of a right chain ordered semigroup (S, ·, ≤). Then for any ideal I of S we have I ⊆ Aor P ( A) ⊆ I. 123 On right chain ordered semigroups 531 Proof Let I be an ideal of S such that P ( A) I . Then there exists p ∈ P ( A)\I . r r Since p ∈ P ( A),for some x ∈ S\ A we have xp ∈ A, and since S is a right chain ordered semigroup and p ∈ / I , it follows that I ⊆ ( pS]. Hence (xS]I ⊆ (xSI]⊆ (xI]⊆ (x ( pS]] ⊆ (xpS]⊆ (AS]⊆ ( A]= A, and since A is prime and x ∈ / A, it follows that I ⊆ A. The following lemma generalizes [9, Lemma 16]. Lemma 3.5 If A is a proper ideal of a right chain ordered semigroup (S, ·, ≤) such 2 n that A = ( A ], then A = (s A] for any s ∈ S\ A and n ∈ N. Proof Let s ∈ S\ A. Since S is a right chain ordered semigroup, we have A ⊆ (sS] and thus A = ( A ]= (AA]⊆ ((sS] A]⊆ (sA]⊆ ( A]= A. Hence A = (sA]. Suppose that for some n ∈ N we have already proved that A = (s A]. Then n n n n+1 A = (s A]= (s (sA]] = (s sA]= (s A]. Thus the result follows by induction. The following two notions are obvious analogues of the concepts deﬁned in [9,p. 580]. Deﬁnition 3.6 Let (S, ·, ≤) be an ordered semigroup. An ideal Q of S is called an exceptional prime ideal of S if Q is prime but not completely prime. If I ⊂ J are ideals of S such that there are no further ideals properly between I and J , then we say that J is minimal over I . We close this section with the following generalization of [9, Lemmas 15 and 17]. Proposition 3.7 Let (S, ·, ≤) be a right chain ordered semigroup, and let Q be an exceptional prime ideal of S. Then there exists a unique ideal D of S such that Q ⊂ D and D is minimal over Q. Furthermore, D = (D ] and there exists an element a ∈ D\Q such that Q ⊂ (a S]. In particular, there exist elements in D\Q that n∈N are not Q-nilpotent. Proof Let D denote the intersection of all ideals I of S such that Q ⊂ I . Proposition 3.4 implies that for any such an ideal I we have P (Q) ⊆ I and thus P (Q) ⊆ D.By r r Proposition 2.5, P (Q) is a completely prime right ideal of S containing Q, and since Q is an exceptional prime ideal, it follows that Q ⊂ P (Q). Hence Q ⊂ D and now the deﬁnition of D and Corollary 2.2 imply that the ideal D is minimal over Q, and 2 2 obviously D is a unique ideal of S with this property. If D = (D ], then (D ]⊂ D, 2 2 2 and the minimality of D over Q implies (D ]⊆ Q. Hence D ⊆ (D ]⊆ Q, and 123 532 T. Changphas et al. since Q is prime, we get D ⊆ Q. This contradiction shows that D = (D ].Wenow prove that there exists a ∈ D\Q such that Q ⊂ (a S]. Set n∈N C ={c ∈ S : (c D]⊆ Q}. n∈N Note that Q ⊆ C and thus the set C is nonempty. We claim that C ⊆ D. Indeed, if s ∈ S\D, then Lemma 3.5 implies that (s D]= D, and since D Q,it n∈N follows that s ∈ / C, which proves our claim. Since Q is an exceptional prime ideal of S, by Proposition 3.1(2) for some b ∈ S\Q we have b ∈ Q.If b ∈ (CbD], then b ∈ (cb D] for some c ∈ C. Hence (bD]⊆ (cb D], and thus 2 2 3 (bD]⊆ (cb D]⊆ (c(bD]] ⊆ (c(cb D]] ⊆ (c (bD]] ⊆ (c (cb D]] ⊆ (c (bD]] ⊆ ··· , n n and continuing this way, we obtain for any n ∈ N that (bD]⊆ (c bD]⊆ (c D]. Hence, since c ∈ C, we obtain (bS]D ⊆ (bS D]⊆ (bD]⊆ (c D]⊆ Q, n∈N and since Q is prime, b ∈ Q or D ⊆ Q, a contradiction. Hence b ∈ / (CbD], and since S is a right chain ordered semigroup, we must have (CbD]⊆ (bS].If C = D, then (DbD]⊆ (bS] and thus 2 2 2 ((bS]D] ⊆ (bD] = (bD](bD]⊆ (bDbD]= (b(DbD]] ⊆ (b(bS]] ⊆ (b S]⊆ Q. Since Q is prime and ((bS]D] ⊆ Q, we get (bS]D ⊆ Q,so b ∈ Q or D ⊆ Q,a contradiction. Hence we must have C ⊂ D. To complete the proof, take any a ∈ D\C. Then (a D] Q, and since S is a right chain ordered semigroup, we obtain n∈N n n Q ⊂ (a D]⊆ (a S]. n∈N n∈N 4 Prime segments of right chain ordered semigroups Following [9], we deﬁne a prime segment of a right chain ordered semigroup (S, ·, ≤) to be a pair P ⊂ P of completely prime ideals of S such that no further completely 2 1 prime ideal of S exists between P and P . In the following theorem we extend to 2 1 right chain ordered semigroups the classiﬁcation of prime segments of right chain semigroups given in [9, Theorem 18]. Theorem 4.1 Let (S, ·, ≤) be a right chain ordered semigroup, and let P ⊂ P be a 2 1 prime segment of S. Then one of the following possibilities occurs: (a) There are no further ideals of S between P and P ; the prime segment is called 2 1 simple in this case. 123 On right chain ordered semigroups 533 (b) For every a ∈ P \ P there exists an ideal I ⊆ P of S such that a ∈ I and 1 2 1 (I ]= P ; the prime segment is called archimedean in this case. n∈N (c) There exists a prime ideal Q of S with P ⊂ Q ⊂ P ; the prime segment is called 2 1 exceptional in this case. (d) There exists an ideal D of S such that P ⊂ D ⊂ P and D is minimal over P ; 2 1 2 the prime segment is called supplementary in this case. Possibilities (a), (b), (c) are mutually exclusive, and possibilities (a), (b), (d) are mutu- ally exclusive. Proof Assume that the prime segment P ⊂ P is not simple, i.e., case (a) does not 2 1 hold. Then there exists an ideal I of S such that P ⊂ I ⊂ P .If P H (S), 2 1 1 I then since S is a right chain ordered semigroup, we must have H (S) ⊂ P , and by I 1 combining Theorem 2.4(1,3) with Proposition 3.1(1) we can see that H (S) is a prime ideal of S lying properly between P and P . Thus the prime segment P ⊂ P is 2 1 2 1 exceptional in this case. Hence to the end of the proof we assume that there exists an ideal I of S with P ⊂ I ⊂ P and for any such an ideal I 2 1 we have P ⊆ H (S). (4.1) 1 I Let us ﬁrst consider the case where the prime segment P ⊂ P contains an ideal 2 1 m m+1 I of S such that (I ]= (I ] for some m ∈ N. Then m+1 m m m+1 m+2 (I ]= (I I]= ((I ]I]= ((I ]I]= (I ] m m+1 m+2 and thus (I ]= (I ]= (I ]. Continuing this way we obtain m m+k (I ]= (I ] for any k ∈ N. m m mn m n Thus for the ideal D = (I ] and any n ∈ N we have D = (I ]= (I ]= ((I ] ]= n m (D ] and D ⊆ I ⊂ P . If we would have D = (I ]⊆ P , then since P is completely 1 2 2 prime, we would get I ⊆ P ⊂ I , a contradiction. Hence, since S is a right chain ordered semigroup, we must have P ⊂ D. We show that furthermore D is minimal over P . If not, then there exists an ideal A of S such that P ⊂ A ⊂ D. Then 2 2 P ⊂ A ⊂ P and by (4.1)wehave D ⊂ P ⊆ H (S). Hence by Proposition 3.2(1), 2 1 1 A (D ]= D is a completely prime ideal of S, which however is impossible, since n∈N P ⊂ P is a prime segment. Hence D is minimal over P and thus the prime segment 2 1 2 P ⊂ P is supplementary in this case. 2 1 We are left with the case where there exists an ideal I of S such that P ⊂ I ⊂ P 2 1 n n+1 and for any such an ideal I we have (I ] = (I ] for all n ∈ N.Let I be the set of all ideals I of S such that P ⊂ I ⊂ P . Since S is a right chain ordered semigroup and 2 1 the ideal P is completely prime, for any I ∈ I and n ∈ N we have P ⊆ (I ], and 2 2 n n thus P ⊆ (I ]⊂ P . Since by Corollary 3.3 the ideal (I ] is completely 2 1 n∈N n∈N prime, it follows that 123 534 T. Changphas et al. (I ]= P for any I ∈ I. (4.2) n∈N Let Q = {I : I ∈ I}. By Corollary 2.2, Q is an ideal of S.If Q = P , then (4.2) implies that the prime segment P ⊂ P is archimedean. Assume that Q = P . Then 2 1 1 Q ⊂ P . We consider two cases: 2 2 2 Case1 : (P ]⊂ P . Then P ⊂ ( P ]⊂ P and thus ( P ]∈ I. Hence 1 2 1 1 1 1 2n 2 n ( P ]= (( P ] ]= P by (4.2) and thus n∈N 1 n∈N 1 n 2n P ⊆ ( P ]⊆ ( P ]⊆ P . 2 2 1 1 n∈N n∈N Therefore P = ( P ] and the prime segment P ⊂ P is archimedean in this 2 2 1 n∈N 1 case. Case2 : (P ]= P . We show that the ideal Q is prime in this case. By Proposition 3.1(1), it sufﬁces to show that Q is semiprime. For this, consider any right ideal A 2 2 of S such that A ⊆ Q. Then A ⊆ P , and since P is completely prime, A ⊆ P 1 1 1 2 2 follows. If A = P , then P = ( P ]= ( A ]⊆ (Q]= Q ⊂ P , a contradiction. 1 1 1 Hence A ⊂ P and thus A ⊆ Q by the deﬁnition of Q. Therefore Q is prime and the prime segment P ⊂ P is exceptional in this case. 2 1 It is easy to see that possibilities (a), (b), (c) are mutually exclusive. It is also clear that (a) and (d) are mutually exclusive. To complete the proof, assume that the possibility (d) occurs and D is minimal over P . Then D = (D ], and Lemma 3.5 n n implies that for any a ∈ P \D we have P ⊂ D = (a D]⊆ (a S]. 1 2 n∈N n∈N Hence (b) and (d) cannot happen simultaneously. Example 19 from [9] shows that possibilities (c) and (d) of Theorem 4.1 can occur simultaneously. We close the paper with the following characterization of archimedean prime seg- ments of right chain ordered semigroups. The result is a generalization of [9, Corollary 20]. Corollary 4.2 Let P ⊂ P be a prime segment of a right chain ordered semigroup 2 1 (S, ·, ≤). Then the following conditions are equivalent. (i) The prime segment P ⊂ P is archimedean. 2 1 (ii) For any a ∈ P \ P , (a S]= P . 1 2 2 n∈N (iii) For any a ∈ P \ P , ( P aS]⊂ (aS]. 1 2 1 Proof (i) ⇒ (ii) follows directly from the deﬁnition of an archimedean prime segment. (ii) ⇒ (iii) Let a ∈ P \ P . Suppose that (aS]⊆ ( P aS]. Then a ≤ pas for some 1 2 1 p ∈ P and s ∈ S.If p ∈ P , then a ∈ ( P aS]⊆ ( P ]= P , a contradiction. Hence 1 2 2 2 2 p ∈ P \P . Furthermore, a ≤ pas implies 1 2 2 2 2 2 3 3 3 3 4 4 a ≤ pas ≤ p( pas)s = p as ≤ p ( pas)s = p as ≤ p ( pas)s = p as ≤ ... n n n and thus for any n ∈ N we have a ≤ p as , and a ∈ ( p S] follows. Hence by (ii), a ∈ ( p S]= P , which is a contradiction. Thus (aS] ( P aS], and since S is 2 1 n∈N a right chain ordered semigroup, we obtain ( P aS]⊂ (aS], as desired. 123 On right chain ordered semigroups 535 (iii) ⇒ (i) Assume (iii). Then for any a ∈ P \ P we have P ⊂ ( P aS]⊂ 1 2 2 1 (aS]⊆ P and thus the prime segment P ⊂ P is not simple. Suppose the prime 1 2 1 segment P ⊂ P is exceptional, i.e., there exists a prime ideal Q of S such that 2 1 P ⊂ Q ⊂ P . Then by Proposition 3.7 there exists an ideal D of S which is minimal 2 1 over Q. This however is impossible, since (iii) implies that for any a ∈ D\Q we have Q ⊂ ( P aS]⊂ (aS]⊆ D. Finally, suppose the prime segment P ⊂ P is 1 2 1 supplementary. Then there exists an ideal D of S such that P ⊂ D ⊂ P and 2 1 D is minimal over P . Then by (iii), for any a ∈ D \ P we have P ⊆ ( P aS]⊂ 2 2 2 1 (aS]⊆ D , a contradiction. Hence the prime segment P ⊂ P is neither simple, nor 2 1 exceptional, nor supplementary, and thus by Theorem 4.1 it must be archimedean. Acknowledgements The authors are grateful to Professor Mikhail Volkov for help in preparing the paper. P. Luangchaisri was supported by Research Fund for Supporting Lecturer to Admit High Potential Student to Study and Research on His Expert Program Year 2016. 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