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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On semisimple semigroup rings


Authors: Mark L. Teply, E. Geis Turman and Antonio Quesada
Journal: Proc. Amer. Math. Soc. 79 (1980), 157-163
MSC: Primary 20M25; Secondary 16A12
MathSciNet review: 565329
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Abstract: Let $ \pi $ be a property of rings that satisfies the conditions that (i) homomorphic images of $ \pi $-rings are $ \pi $-rings and (ii) ideals of $ \pi $-rings are $ \pi $-rings. Let S be a semilattice P of semigroups $ {S_\alpha }$. If each semigroup ring $ R[{S_\alpha }](\alpha \in P)$ is $ \pi $-semisimple, then the semigroup ring $ R[{S_\alpha }]$ is also $ \pi $-semisimple. Conditions are found on P to insure that each $ R[{S_\alpha }](\alpha \in P)$ is $ \pi $-semisimple whenever S is a strong semilattice P of semigroups $ {S_\alpha }$ and $ R[S]$ is $ \pi $-semisimple. Examples are given to show that the conditions on P cannot be removed. These results and examples answer several questions raised by J. Weissglass.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1980-0565329-2
PII: S 0002-9939(1980)0565329-2
Keywords: Semigroup ring, semilattice, $ \pi $-semisimple
Article copyright: © Copyright 1980 American Mathematical Society