Commutative Rings Whose proper Ideals are ℘℘-Virtually Semisimple

This paper is a continuation of our previous article (Behbood and Bigdeli in Commun Algebra 47:3995–4008, 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 idealM= Soc(R), or a local ring with maximal idealM, such thatM∼= Soc(R) ⊕ ( λ∈ R/Pλ) where is an index set and {Pλ|λ ∈ } is the set of all non-maximal prime ideals of R, and for each Pλ, the ring R/Pλ is a principal ideal domain.We also characterize commutative rings R whose proper ideals are ℘-virtually semisimple. It is shown that every proper ideal of R is ℘-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 idealM= Soc(R), or a local ring with maximal idealM ∼= Soc(R) ⊕ R/P, where P is the only non-maximal prime ideal of R and R/P is a principal ideal domain.