Modules with epimorphisms on chains of submodules

摘要

We say that an R-module M satisfies epi-ACC on submodules if in every ascending chain of submodules of M, except probably a finite number, each module in chain is a homomorphic image of the next one. Noetherian modules, semisimple modules and Prufer p-groups have this property. Direct sums of modules with epi-ACC on submodules need not have this property. If E(R-R)((N)) satisfies epi-ACC on submodules, then R is quasi-Frobenius. As a consequence, a ring R in which all modules satisfy epi-ACC on submodules is an artinian principal ideal ring. Dually, we say that an R-module M satisfies epi-DCC on submodules if in every descending chain of submodules of M, 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 R for which all modules satisfy epi-DCC on submodules must be an artinian principal ideal ring.