A ring R is called right א0-injective if every right homomorphism from a countably generated right ideal of R to RR can be extended to a homomorphism from RR to RR. In this note, some characterizations of א0-injective rings are given. It is proved that if R is semiperfect, then R is right א0injective if and only if every homomorphism from a countably generated small right ideal of R to RR can b...

It is well-known that any semiperfect A ring has a decomposition as a direct sum (product) of indecomposable subrings A = A1 ⊕ · · · ⊕ An such that the Ai-Mod are indecomposable module categories. Similarly any coalgebra C over a field can be written as a direct sum of indecomposable subcoalgebras C = ⊕ I Ci such that the categories of Ci-comodules are indecomposable. In this paper a decomposit...

The theory of modules over associative algebras and the theory of comodules for coassociative coalgebras were developed fairly independently during the last decades. In this survey we display an intimate connection between these areas by the notion of categories subgenerated by an object. After a review of the relevant techniques in categories of left modules, applications to the bimodule struc...

Our investigation of coalgebras over commutative rings R is based on the close relationship between comodules over a coalgebra C and modules over the dual algebra C∗. If C is projective as an R-module the category of right C-comodules can be identified with the category σ[C∗C] of left C∗-modules which are subgenerated by C. In this context semiperfect coalgebras are described by results from mo...

We investigate two invariants of Noetherian semiperfect rings, namely the depth and a new invariant we call “delooping level”. These give lower upper bounds for finitistic dimension, respectively. As first theorems, necessary sufficient criterion delooping level to be finite in terms splitting unit map related adjunction, use this torsionfreeness finiteness. further relate these Auslander–Bridg...

We give a characterization of right Noetherian semiprime semiperfect and semidistributive rings with inj. dimAAA 6 1.