We say a left R-module M is ecg-extending if every essentially countably generated submodule of M is essential in a direct summand of M. After giving some basic properties of ecg-extending modules, we show that for a nonsingular ring R, all left R-modules are ecg- extending if and only if all left R-modules are extending. We also characterize noetherian rings and artinian semisimple rings via ecg-quasi-continuous modules.
Let R be a ring. Recall that a right R-module M (RR, resp.) is said to be a PS-module (PS-ring, resp.) if it has projective socle. M is called a CESS-module if every complement summand in M with essential socle is a direct summand of M. We show that the formal triangular matrix ring T = A 0M B is a PS-ring if and only if A is a PS-ring, MA and lB(M) = {b ∈ B | bm = 0,m ∈ M} are PS-modules and Soc(lB(M)) M = 0. Using the alternative of right T-module as triple (X,Y )f with X ∈ Mod-A, Y ∈ Mod-B and f : YM →...
This paper is motivated by S. Park [10] in which the injective cover of left R[x]- module M[x? ] of inverse polynomials over a left R-module M was discussed. The 1 author considers the ?-covers of modules and shows that if η : P ?→ M is an ?- cover of M, then [ηS, ] : [PS, ] ?→ [MS, ] is an [?S, ]-cover of left [[RS, ]]-module ≤ ≤ ≤ ≤ ≤ [MS, ], where ? is a class of left R-modules and [MS, ] is the left [[RS, ]]-module of ≤ ≤ ≤ generalized inverse polynomials over a left R-module M. Also some properties of the injective cover of left [[RS, ]]-module [MS, ] are discussed. ≤
Let R be a ring such that all left semicentral idempotents axe central and α a weakly rigid endomorphism of R. It is shown that the skew power series ring R[[x; α]] is right p.q.Baer if and only if R is right p.q.Baer and any countable family of idempotents in R has a generalized join in I(R), where I(R) is the set of all idempotents of R.
