Journal of Algebra and Related Topics
Volume:4 Issue: 1, Summer 2016

et \pounds be the category of locally compact abelian groups and A,C∈\pounds. In this paper, we define component extensions of A by C and show that the set of all component extensions of A by C forms a subgroup of Ext(C,A) whenever A is a connected group. We establish conditions under which the component extensions split and determine LCA groups which are component projective. We also gives a necessary condition for an LCA group to be component injective in \pounds.

Deleanu, Frei, and Hilton have developed the notion of generalized Adams completion in a categorical context. In this paper, we have obtained the Postnikov-like approximation of a module, with the help of a suitable set of morphisms.

Let R be a commutative ring and let M be an R-module. We define the small intersection graph G(M) of M with all non-small proper submodules of M as vertices and two distinct vertices N,K are adjacent if and only if N∩K is a non-small submodule of M. In this article, we investigate the interplay between the graph-theoretic properties of G(M) and algebraic properties of M, where Mis a multiplication module.

Let (R,m) be a commutative Noetherian local ring, M a finitely generated R-module of dimension d, and let I be an ideal of definition for M. In this paper, we extend \cite[Corollary 10(4)]{P} and also we show that if M is a Cohen-Macaulay R-module and d=2, then λ(InM˜JIn−1M˜) does not depend on J for all n≥1, where J is a minimal reduction of I.

For an arbitrary ring R, the zero-divisor graph of R, denoted by Γ(R), is an undirected simple graph that its vertices are all nonzero zero-divisors of R in which any two vertices x and y are adjacent if and only if either xy=0 or yx=0. It is well-known that for any commutative ring R, Γ(R)≅Γ(T(R)) where T(R) is the (total) quotient ring of R
. In this paper we extend this fact for certain noncommutative rings, for example, reduced rings, right (left) self-injective rings and one-sided Artinian rings. The necessary and sufficient conditions for two reduced right Goldie rings to have isomorphic zero-divisor graphs is given. Also, we extend some known results about the zero-divisor graphs from the commutative to noncommutative setting: in particular, complemented and uniquely complemented graphs.

In this paper we define an order structure on the p-operator projective tensor product of Herz algebras and we show that the canonical isometric isomorphism between Ap(G×H) and Ap(G)⊗ˆpAp(H) is an order isomorphism for amenable groups G and H.