Communications in Algebra

Banerjee, Abhishek

doi: 10.1080/00927872.2011.600399pmid: N/A

Given a monoid object A in a symmetric monoidal category (C, ⊗, 1), we associate a commutative monoid z(A): = Hom A−Bimod (A, A), which we refer to as the set of central elements of A. If A has a centre Z(A) in C, we describe an isomorphism z(A) = Hom A−Bimod (A, A) ≅ Hom Z(A)−Mod (Z(A), Z(A)). When C is closed and complete, we show that every subset S ⊆ Hom A−Mod (A, A) has a centraliser Z A (S) which is a Z(A)-algebra and under certain conditions, we show that Z A (Hom A−Mod (A, A)) = Z(A). Further, for any multiplicatively closed set S ⊆ z(A), we define the localisation M S of any right A-module M with respect to S and show that several properties of classical localisation extend to this context.

He, Liguo; Qian, Guohua

doi: 10.1080/00927872.2011.600400pmid: N/A

In this note, we classify the finite groups admitting a nonsolvable normal subgroup with three induced irreducible character degrees.

Habibi, M.; Moussavi, A.; Mokhtari, S.

doi: 10.1080/00927872.2011.600746pmid: N/A

Let α be an automorphism of a ring R. We study the skew Armendariz of Laurent series type rings (α-LA rings), as a generalization of the standard Armendariz condition from polynomials to skew Laurent series. We study on the relationship between the Baerness and p.p. property of a ring R and these of the skew Laurent series ring R[[x, x −1; α]], in case R is an α-LA ring. Moreover, we prove that for an α-weakly rigid ring R, R[[x, x −1; α]] is a left p.q.-Baer ring if and only if R is left p.q.-Baer and every countable subset of S ℓ(R) has a generalized countable join in R. Various types of examples of α-LA rings are provided.

Sedaghatjoo, M.; Laan, V.; Ershad, M.

doi: 10.1080/00927872.2011.600747pmid: N/A

For a monoid S, the set S × S equipped with the componentwise right S-action is called the diagonal act of S and is denoted by D(S). A monoid S is a left PP (left PSF) monoid if every principal left ideal of S is projective (strongly flat). We shall call a monoid S left P(P) if all principal left ideals of S satisfy condition (P). We shall call a monoid S weakly left P(P) monoid if the equalities as = bs, xb = yb in S imply the existence of r ∈ S such that xar = yar, rs = s. In this article, we prove that a monoid S is left PSF if and only if S is (weakly) left P(P) and D(S) is principally weakly flat. We provide examples showing that the implications left PSF ⇒ left P(P) ⇒ weakly left P(P) are strict. Finally, we investigate regularity of diagonal acts D(S), and we prove that for a right PP monoid S the diagonal act D(S) is regular if and only if every finite product of regular acts is regular. Furthermore, we prove that for a full transformation monoid S = 𝒯 X , D(S) is regular.

Hajja, Mowaffaq

doi: 10.1080/00927872.2011.601485pmid: N/A

Let R be a commutative ring with identity, and let R[x] be the polynomial ring in one indeterminate x over R. We give algorithms for calculating σ−1(x), where σ is a given R-automorphism of R[x], thus answering a question raised in [2]. Needless to say, this problem is trivial for nonreduced rings R, since an R-automorphism σ of R[x] in this case must have the simple form σ(x) = ux + c, where u, c ∈ R and where u is a unit.

Demir, Ç.; Albaş, E.; Argaç, N.; De Filippis, V.

doi: 10.1080/00927872.2012.689393pmid: N/A

Let R be a prime ring, f(X 1, …, X n ) a multilinear polynomial which is not central-valued on R, and G a nonzero generalized skew derivation of R. Suppose that G(f(x 1, …, x n )) is zero or invertible for all x 1, …, x n  ∈ R. Then it is proved that R is either a division ring or the ring of all 2 × 2 matrices over a division ring. This result simultaneously generalizes a number of results in the literature.

Singla, Pooja

doi: 10.1080/00927872.2011.601240pmid: N/A

We study the complex irreducible representations of special linear, symplectic, orthogonal, and unitary groups over principal ideal local rings of length two. We construct a canonical correspondence between the irreducible representations of all such groups that preserves dimensions. The case for general linear groups has already been proved by the author.

Brunshidle, Lindsay; Fialowski, Alice; Frinak, Josh; Penkava, Michael; Wackwitz, Dan

doi: 10.1080/00927872.2011.602162pmid: N/A

In this article, we give an extension of the Fundamental Theorem of finite dimensional algebras to the case of ℤ2-graded algebras. Essentially, the results are the same as in the classical case, except that the notion of a ℤ2-graded division algebra needs to be modified. We classify all finite dimensional ℤ2-graded division algebras over ℂ and ℝ.

Alan, Murat

doi: 10.1080/00927872.2011.602163pmid: N/A

Let R be a commutative ring with identity. R is a finite factorization ring (FFR) if every nonzero nonunit of R has only a finite number of factorizations up to order and associates. In this article, we give a characterization of R for R[X] and R[[X]] to be an FFR.

Kour, Surjeet

doi: 10.1080/00927872.2011.602164pmid: N/A

It is shown that the derivation y r ∂ x  + (xy s  + g)∂ y , where 0 ≤ r < s are integers, is a simple derivation of k[x, y], the polynomial ring in two variables over a field k of characteristic zero.