Weakly-morphic modules

Characterizations of regular modules

“…Using this assumption and the First Isomorphism Theorem for the R-endomorphism φ : R → S := End R (M ) defined by φ(a) = φ a for all a ∈ R, we obtain {φ a : a ∈ R} is regular. By [13,Proposition 7], M is a weakly-endoregular module.…”

“…This notion has been widely studied, see for instance [21]. Recently for commutative rings, M is called weakly-morphic in [13] if M/M a ∼ = l M (a) as R-modules for each a ∈ R, i.e., if every endomorphism φ a of M given by right multiplication by a ∈ R is morphic. It turns out that a (commutative) ring R is right (and left) morphic if and only if the R-module R is a weakly-morphic module.…”

Characterizations of regular modules

“…This notion has been widely studied, see for instance [21]. Recently for commutative rings, M is called weakly-morphic in [13] if M/Ma ∼ = l M (a) as R-modules for each a ∈ R, i.e., if every endomorphism ϕ a of M given by right multiplication by a ∈ R is morphic. It turns out that a (commutative) ring R is right (and left) morphic if and only if the R-module R is a weakly-morphic module.…”

“…Using this assumption and the First Isomorphism Theorem for the R-endomorphism ϕ : R → S := End R (M) defined by ϕ(a) = ϕ a for all a ∈ R, we obtain {ϕ a : a ∈ R} is regular. By [13,Proposition 7], M is a weakly-endoregular module.…”

“…Lemma 6 [13,. Proposition 19] Let M be a finitely generated multiplication module over a commutative ring R. Then M is weakly-morphic if and only if it is morphic.…”