By any measure, semisimple modules form one of the most important classes of modules and play a distinguished role in the module theory and its applications. One of the most fundamental results in this area is the Wedderburn-Artin theorem. In this paper, we establish natural generalizations of semisimple modules and give a generalization of the Wedderburn-Artin theorem. We study modules in which every submodule is isomorphic to a direct summand and name them virtually semisimple modules. A module R M is called completely virtually semisimple if each submodules of M is a virtually semisimple module. A ring R is then called left (completely) virtually semisimple if R R is a left (compleatly) virtually semisimple R-module. Among other things, we give several characterizations of left (completely) virtually semisimple rings. For instance, it is shown that a ring R is left completely virtually semisimple if and only ifwhere k, n 1 , ..., n k ∈ N and each D i is a principal left ideal domain. Moreover, the integers k, n 1 , ..., n k and the principal left ideal domains D 1 , ..., D k are uniquely determined (up to isomorphism) by R. * The research of the first author was in part supported by a grant from IPM (No. 94130413). This research is partially carried out in the IPM-Isfahan Branch.†
We say that an R-module M is virtually simple if M = (0) and N ∼ = M for every nonzero submodule N of M , and virtually semisimple if each submodule of M is isomorphic to a direct summand of M . We carry out a study of virtually semisimple modules and modules which are direct sums of virtually simple modules. Our theory provides two natural generalizations of the Wedderburn-Artin Theorem and an analogous to the classical Krull-Schmidt Theorem. Some applications of these theorems are indicated. For instance, it is shown that the following statements are equivalent for a ring R: (i) Every finitely generated left (right) R-modules is virtually semisimple; (ii) Every finitely generated left (right) R-modules is a direct sum of virtually simple modules;n 1 , . . . , n k ∈ N and each D i is a principal ideal V-domain; and (iv) Every non-zero finitely generated left R-module can be written uniquely (up to isomorphism and order of the factors) in the form Rm 1 ⊕ . . . ⊕ Rm k where each Rm i is either a simple R-module or a left virtually simple direct summand of R. * The research of the first author was in part supported by a grant from IPM (No. 95130413). This research is partially carried out in the IPM-Isfahan Branch. † Key Words: Virtually simple module; virtually semisimple module; left principal ideal domain; Vdomain; FGC-ring; Wedderburn-Artin Theorem; Krull-Schmidt Theorem.
A restricted artinian ring is a commutative ring with an identity in which every proper homomorphic image is artinian. Cohen proved that a commutative ring R is restricted artinian if and only if it is noetherian and every nonzero prime ideal of R is maximal. Facchini and Nazemian called a commutative ring isoartinian if every descending chain of ideals becomes stationary up to isomorphism. We show that every proper homomorphic image of a commutative noetherian ring R is isoartinian if and only if R has one of the following forms: (a) R is a noetherian domain of Krull dimension one which is not a principal ideal domain; (b) R ≅ D 1 × ⋯ × D k × A 1 × ⋯ × A l {R\cong D_{1}\times\cdots\times D_{k}\times A_{1}\times\cdots\times A_{l}} , where each D i {D_{i}} is a principal ideal domain and each A i {A_{i}} is an artinian local ring (either k or l may be zero); (c) R is a noetherian ring of Krull dimension one, simple unique minimal prime ideal 𝔭 {\mathfrak{p}} , and R / 𝔭 {R/\mathfrak{p}} is a principal ideal domain. As an application of our result, we describe commutative rings whose proper homomorphic images are principal ideal rings. Some relevant examples are provided.
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