Virtually semisimple modules and a generalization of the Wedderburn-Artin theorem. (English) Zbl 1430.16008
Summary: 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.
MSC:
| 16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |
| 13F10 | Principal ideal rings |
| 16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |
| 16S50 | Endomorphism rings; matrix rings |
Keywords:
completely virtually semisimple ring; principal left ideal ring; semisimple module; semisimple ring; virtually semisimple ring; Wedderburn-Artin theoremReferences:
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