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Paper   IPM / M / 16957
School of Mathematics
  Title:   Commutative rings whose proper ideals are P-virtually semisimple
  Author(s):  Mahmood Behboodi (Joint wuth E. Bigdeli)
  Status:   Published
  Journal: Bull. Iranian Math. Soc.
  Vol.:  48
  Year:  2022
  Pages:   1-12
  Supported by:  IPM
  Abstract:
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 ideal M = Soc(R), or a local ring with maximal ideal M, such that M ∼= 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 ideal M = Soc(R), or a local ring with maximal ideal M ∼= Soc(R) ⊕ 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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