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This paper is a continuation of our previous article (Behbood and Bigdeli in Commun Algebra 47:3995â4008, 2019). We study commutative rings R whose proper (prime) ideals are direct sums of virtually simple R-modules. It is shown that every prime ideal of R is a direct sum of virtually simple R-modules, if and only if either R is a finite direct product of principal ideal domains, a local ring with maximal ideal M = Soc(R), or a local ring with maximal ideal M, such that M â¼= Soc(R) â (Î»â R/PÎ») where
is an index set and {PÎ»|Î» â } is the set of all non-maximal prime ideals of R, and for each PÎ», the ring R/PÎ» is a principal ideal domain. We also characterize commutative rings R whose proper ideals are â-virtually semisimple. It is shown that every proper ideal of R is â-virtually semisimple if and only if every proper ideal of R is a direct sum of virtually simple R-modules, if and only if either R is a finite direct product of principal ideal domains, a local ring with maximal ideal M = Soc(R), or a local ring with maximal ideal M â¼= Soc(R) â R/P, where P is the only non-maximal prime ideal of R and R/P is a principal ideal domain.
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