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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Group rings, matrix rings, and polynomial identities


Author: Elizabeth Berman
Journal: Trans. Amer. Math. Soc. 172 (1972), 241-248
MSC: Primary 16A38
MathSciNet review: 0308184
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Abstract: This paper studies the question, if $ R$ is a ring satisfying a polynomial identity, what polynomial identities are satisfied by group rings and matrix rings over $ R$? Theorem 2.6. If $ R$ is an algebra over a field with at least $ q$ elements, and $ R$ satisfies $ {x^q} = 0$, and $ G$ is a group with an abelian subgroup of index $ k$, then the group ring $ R(G)$ satisfies $ {x^t} = 0$, where $ t = q{k^2} + 2$. Theorem 3.2. If $ R$ is a ring satisfying a standard identity, and $ G$ is a finite group, then $ R(G)$ satisfies a standard identity. Theorem 3.4. If $ R$ is an algebra over a field, and $ R$ satisfies a standard identity, then the $ k$-by-$ k$ matrix ring $ {R_k}$ satisfies a standard identity. Each theorem specifies the degree of the polynomial identity.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0308184-8
PII: S 0002-9947(1972)0308184-8
Keywords: Group rings, matrix rings, polynomial identities, standard identity, bounded nil rings
Article copyright: © Copyright 1972 American Mathematical Society