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Paper IPM / M / 16584  


Abstract:  
We study the class of virtually homouniserial modules and rings as a nontrivial generalization of homouniserial modules and rings. An Rmodule M is virtually homouniserial if, for any finitely generated submodules 0 6= K, L ï¿½?? M, the factor modules K/Rad(K) and
L/Rad(L) are virtually simple and isomorphic (an Rmodule M is virtually simple if, M 6= 0 and M ï¿½?ï¿½= N for every nonzero submodule N of M). Also, an Rmodule M is called virtually homoserial if it is a direct sum of virtually homouniserial modules. We obtain that
every left Rmodule is virtually homoserial if and only if R is an Artinian principal ideal ring. Also, it is shown that over a commutative ring R, every finitely generated Rmodule
is virtually homoserial 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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