Modules with epimorphisms on chains of submodules

Abstract

We say that an [Formula: see text]-module [Formula: see text] satisfies epi-ACC on submodules if in every ascending chain of submodules of [Formula: see text], except probably a finite number, each module in chain is a homomorphic image of the next one. Noetherian modules, semisimple modules and Prüfer [Formula: see text]-groups have this property. Direct sums of modules with epi-ACC on submodules need not have this property. If [Formula: see text] satisfies epi-ACC on submodules, then [Formula: see text] is quasi-Frobenius. As a consequence, a ring [Formula: see text] in which all modules satisfy epi-ACC on submodules is an artinian principal ideal ring. Dually, we say that an [Formula: see text]-module [Formula: see text] satisfies epi-DCC on submodules if in every descending chain of submodules of [Formula: see text], except probably a finite number, each module in chain is a homomorphic image of the preceding. Artinian modules, semisimple modules and free modules over commutative principal ideal domains are examples of such modules. A semiprime right Goldie ring satisfies epi-DCC on right ideals if and only if it is a finite product of full matrix rings over principal right ideal domains. A ring [Formula: see text] for which all modules satisfy epi-DCC on submodules must be an artinian principal ideal ring.