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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Matrix rings over polynomial identity rings


Author: Elizabeth Berman
Journal: Trans. Amer. Math. Soc. 172 (1972), 231-239
MSC: Primary 16A42
DOI: https://doi.org/10.1090/S0002-9947-1972-0308187-3
MathSciNet review: 0308187
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Abstract: We prove that if $ A$ is an algebra over a field with at least $ k$ elements, and $ A$ satisfies $ {x^k} = 0$, then $ {A_n}$, the ring of $ n$-by-$ n$ matrices over $ A$, satisfies $ {x^q} = 0$, where $ q = k{n^2} + 1$. Theorem 1.3 generalizes this result to rings: If $ A$ is a ring satisfying $ {x^k} = 0$, then for all $ n$, there exists $ q$ such that $ {A_n}$ satisfies $ {x^q} = 0$.

Definitions. A checkered permutation of the first $ n$ positive integers is a permutation of them sending even integers into even integers. The docile polynomial of degree $ n$ is

$\displaystyle \prod\limits_{i = 1}^p {D({x_{i1}}, \cdots ,{x_{ik}}){u_i},} $

athewhere the sum is over all checkered permutations $ f$ of the first $ k$ positive integers. The docile product polynomial of degree $ k,p$is

$\displaystyle \prod\limits_{i = 1}^p {D({x_{i1}}, \cdots ,{x_{ik}}){u_i},} $

where the $ x$'s and $ u$'s are noncommuting variables. Theorem 2.1. Any polynomial identity algebra over a field of characteristic 0 satisfies a docile product polynomial identity. Theorem 2.2. If $ A$ is a ring satisfying the docile product polynomial identity of degree $ 2k,p$, and $ n$ is a positive integer, and $ q = 2{k^2}{n^2} + 1$; then $ {A_n}$ satisfies a product of $ p$ standard identities, each of degree $ q$.

References [Enhancements On Off] (What's this?)

  • [1] Elizabeth Berman, Tensor products of polynomial identity algebras, Trans. Amer. Math. Soc. 156 (1971), 259-271. MR 43 #278. MR 0274515 (43:278)
  • [2] Nathan Jacobson, Structure of rings, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 37, Amer. Math. Soc., Providence, R. I., 1964. MR 36 #5158. MR 0222106 (36:5158)
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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0308187-3
Keywords: Polynomial identity rings, nil rings, matrix rings
Article copyright: © Copyright 1972 American Mathematical Society

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