Rings whose modules are weakly supplemented are perfect
Engin B\"uy\"uka\c{s}ik, Christian Lomp
TL;DR
This paper characterizes left perfect rings through the property that all left modules are weakly supplemented, linking ring perfection to module-theoretic conditions involving semilocality and weak supplements.
Contribution
It establishes a new equivalence for left perfect rings based on weakly supplemented modules and properties of free modules.
Findings
A ring R is left perfect iff every left R-module is weakly supplemented.
Left perfect rings are characterized by semilocality and weak supplements of certain free modules.
The radical of the countably infinite free left R-module has a weak supplement in left perfect rings.
Abstract
In this note we show that a ring R is left perfect if and only if every left R-module is weakly supplemented if and only if R is semilocal and the radical of the countably infinite free left R-module has a weak supplement.
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Taxonomy
TopicsRings, Modules, and Algebras · Advanced Topics in Algebra · Advanced Algebra and Logic