“School of Mathematics”
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| Paper IPM / M / 16584 |
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| Abstract: | |||||
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We study the class of virtually homo-uniserial modules and rings as a nontrivial generalization of homo-uniserial modules and rings. An R-module M is virtually homo-uniserial if, for any finitely generated submodules 0 6= K, L �?? M, the factor modules K/Rad(K) and
L/Rad(L) are virtually simple and isomorphic (an R-module M is virtually simple if, M 6= 0 and M �?�= N for every non-zero submodule N of M). Also, an R-module M is called virtually homo-serial if it is a direct sum of virtually homo-uniserial modules. We obtain that
every left R-module is virtually homo-serial if and only if R is an Artinian principal ideal ring. Also, it is shown that over a commutative ring R, every finitely generated R-module
is virtually homo-serial if and only if R is a finite direct product of almost maximal uniserial rings and principal ideal domains with zero Jacobson radical. Finally, we obtain some structure theorems for commutative (Noetherian) rings whose every proper ideal is virtually (homo-)serial.
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