Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem

2017/09/28 by Mahmood Behboodi, Asghar Daneshvar, M. R. Vedadi
Mathematics · #Rings, Modules, and Algebras #Commutative Algebra and Its Applications

paper · doi:10.1080/00927872.2017.1384002

Abstract

A widely used result of Wedderburn and Artin states that “every left ideal of a ring R is a direct summand of R if and only if R has a unique decomposition as a finite direct product of matrix rings over division rings.” Motivated by this, we call a module M virtually semisimple if every submodule of M is isomorphic to a direct summand of M and M is called completely virtually semisimple if every submodule of M is virtually semisimple. We show that the left R-module R is completely virtually semisimple if and only if R has a unique decomposition as a finite direct product of matrix rings over principal left ideal domains. This shows that R is completely virtually semisimple on both sides if and only if every finitely generated (left and right) R-module is a direct sum of a singular module and a projective virtually semisimple module. The Wedderburn-Artin theorem follows as a corollary from our result.

Citations

• Submodules of Direct Sums of Compressible Modules
• Modules with chain conditions up to isomorphism
• Artinian dimension and isoradical of modules

Cited by

• Two Generalizations of the Wedderburn-Artin Theorem with Applications

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