Let A be a finite-dimensional, basic, connected algebra over an algebraically closed field. Denote by Γ (A) the Auslander–Reiten quiver of A. We show that A is representation-finite if and only if Γ (A) has at most finitely many vertices lying on oriented cycles and finitely many orbits with respect to the action of the Auslander–Reiten translation. Let K denote a fixed algebraically closed fie...

All rings are commutative with 1 6= 0, and all modules are unital. The purpose of this paper is to investigate the concept of 2-absorbing primary submodules generalizing 2-absorbing primary ideals of rings. Let M be an R-module. A proper submodule N of an R-module M is called a 2-absorbing primary submodule of M if whenever a, b ∈ R and m ∈M and abm ∈ N , then am ∈M -rad(N) or bm ∈M -rad(N) or ...

LetG be a finite p-group,K a field of characteristic p, and J the radical of the group algebra K[G]. We study modular representations using some new results of the theory of extensions of modules. More precisely, we describe the K[G]-modules M such that J2M = 0 and give some properties and isomorphism invariants which allow us to compute the number of isomorphism classes of K[G]-modulesM such t...

It is well-known that a ring R is semiperfect if and only if RR (or RR) is a supplemented module. Considering weak supplements instead of supplements we show that weakly supplemented modules M are semilocal (i.e., M/Rad(M) is semisimple) and that R is a semilocal ring if and only if RR (or RR) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie d...