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| Paper IPM / M / 17187 |
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A theorem due to Warfield states that âa ring R is left serial if and only if every (finitely
generated) projective left R-module is serialâ and a theorem due to Tuganbaev states that
âa ring R is a finite direct product of uniserial Noetherian rings if and only if R is left duo,
and all injective left R-modules are serialâ. Most recently, in our previous paper [Virtually
uniserial modules and rings, J. Algebra (2020), to appear], we introduced and studied the
concept of virtually uniserial modules as a nontrivial generalization of uniserial modules.
We say that an R-module M is virtually uniserial if, for every finitely generated submodule
0 6= K â M, K/Rad(K) is virtually simple (an R-module M is virtually simple if, M 6= 0 and N â¼= M for every non-zero submodule N of M). Also, an R-module M is called virtually serial if it is a direct sum of virtually uniserial modules. The above results of Warfield and Tuganbaev motivated us to study the following questions: âWhich rings have the property that every (finitely generated) projective module is virtually serial?â and âWhich rings have the property that every injective module is virtually serial?â. The goal of this paper is to answer these questions.
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