Communications in Algebra

Baeth, Nicholas R.

doi: 10.1080/00927870802064663pmid: N/A

Given a ring R and a class 𝒞 of R-modules, one can ask whether or not every element of 𝒞 decomposes uniquely as a direct sum of indecomposable elements of 𝒞. If not, one can further ask if it is possible for an element of 𝒞 to decompose both as the direct sum of s indecomposable elements of 𝒞 and as the direct sum of t indecomposable elements of 𝒞 where s ≠ t. In this article, we investigate these questions when R is a two-dimensional analytically normal domain and 𝒞 is the class of finitely generated torsion-free R-modules.

Bardakov, Valerij G.; Bellingeri, Paolo

doi: 10.1080/00927870802068664pmid: N/A

We prove that pure braid groups of closed surfaces are almost-direct products of residually torsion free nilpotent groups and hence residually torsion free nilpotent. As a corollary, we prove also that braid groups on 2 strands of closed surfaces are residually nilpotent.

El Kaoutit, L.

doi: 10.1080/00927870802070314pmid: N/A

We introduce the notion of “bi-monoid” in general monoidal category, generalizing by this the notion of “bialgebra”. In the case of bimodules over a noncommutative algebra, we obtain compatibility conditions between rings and corings whenever both structures admit the same underlying bimodule. Several examples are expounded in this case. We also show that there is a class of right modules over a bi-monoid which is a monoidal category and the forgetful functor to the ground category is a strict monoidal functor.

Bustamante, Juan Carlos; Dionne, Julie; Smith, David

doi: 10.1080/00927870802089892pmid: N/A

We study three different (co)homology theories for a family of pullbacks of algebras that we call oriented. We obtain a Mayer Vietoris long exact sequence of Hochschild and cyclic homology and cohomology groups for these algebras. We give examples showing that our sequence for Hochschild cohomology groups is different from the known ones. In case the algebras are given by quiver and relations, and that the simplicial homology and cohomology groups are defined, we obtain a similar result in a slightly wider context. Finally, we also study the fundamental groups of the bound quivers involved in the pullbacks.

Emtander, Eric

doi: 10.1080/00927870802098158pmid: N/A

In this article, we study some algebraic properties of hypergraphs, in particular their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions.

Aziziheris, Kamal; Lewis, Mark L.

doi: 10.1080/00927870802112884pmid: N/A

Suppose G is a finite group. We show that the number of nonlinear irreducible character of G is at least . In addition, we prove that this bound is obtained if and only if |G′| = 2. We show that this bound can be improved under many different additional hypotheses.

Kutami, Mamoru

doi: 10.1080/00927870802116513pmid: N/A

We study regular rings satisfying generalized almost comparability. First, we determine the forms of regular rings satisfying generalized almost comparability. Next, using the above result, we treat the strict cancellation property and the strict unperforation property for regular rings satisfying generalized almost comparability.

D'Anna, M.; Mezzasalma, M.; Micale, V.

doi: 10.1080/00927870802116521pmid: N/A

Let (R, m) be a Noetherian, one-dimensional, local ring, with |R/m|=∞. We study when its associated graded ring G(m) is Buchsbaum; in particular, we give a theoretical characterization for G(m) to be Buchsbaum not Cohen–Macaulay. Finally, we consider the particular case of R being the semigroup ring associated to a numerical semigroup S: we introduce some invariants of S, and we use them in order to give a necessary and a sufficient condition for G(m) to be Buchsbaum.

Lee, Gregory T.; Spinelli, Ernesto

doi: 10.1080/00927870802116539pmid: N/A

Let K be an infinite field of characteristic different from 2, and G a group. Under suitable restrictions upon G, we classify the groups such that the symmetric units of KG satisfy the solvability identity (x 1, x 2,…, x 2 n ) o  = 1, for some n.

Álvarez, Amelia; Sancho, Carlos; Sancho, Pedro

doi: 10.1080/00927870802116547pmid: N/A

Let R be a commutative ring with unit, and let E be an R-module. We say the functor of R-modules E, defined by E(B) = E ⊗ R B, is a quasi-coherent R-module, and its dual E* is an R-module scheme. Both types of R-module functors are essential for the development of the theory of the linear representations of an affine R-group. We prove that a quasi-coherent R-module E is an R-module scheme if and only if E is a projective R-module of finite type, and, as a consequence, we also characterize finitely generated projective R-modules.