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Transactions of the American Mathematical Society

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



Tensor products of polynomial identity algebras

Author: Elizabeth Berman
Journal: Trans. Amer. Math. Soc. 156 (1971), 259-271
MSC: Primary 16.49
MathSciNet review: 0274515
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Abstract: We investigate matrix algebras and tensor products of associative algebras over a commutative ring R with identity, such that the algebra satisfies a polynomial identity with coefficients in R. We call A a P. I. algebra over R if there exists a positive integer n and a polynomial f in n noncommuting variables with coefficients in R, not annihilating A, such that for all $ {a_1}, \ldots ,{a_n}$ in A, $ f({a_1}, \ldots ,{a_n}) = 0$. We call A a P-algebra if f is homogeneous with at least one coefficient of 1. We define the docile identity, a polynomial identity generalizing commutativity, in that if A satisfies a docile identity, then for all n, $ {A_n}$, the set of n-by-n matrices over A, satisfies a standard identity. We similarly define the unitary identity, which generalizes anticommutativity. Claudio Procesi and Lance Small recently proved that if A is a P. I. algebra over a field, then for all n, $ {A_n}$ satisfies some power of a standard identity. We generalize this result to P-algebras over commutative rings with identity. It follows that if A is a P-algebra, A satisfies a power of the docile identity.

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

  • [1] I. N. Herstein, Noncommutative rings, The Carus Mathematical Monographs, No. 15, Published by The Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968. MR 0227205
  • [2] Nathan Jacobson, Structure of rings, American Mathematical Society Colloquium Publications, Vol. 37. Revised edition, American Mathematical Society, Providence, R.I., 1964. MR 0222106
  • [3] Claudio Procesi and Lance Small, Endomorphism rings of modules over 𝑃𝐼-algebras, Math. Z. 106 (1968), 178–180. MR 0233846

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Keywords: Polynomial identity, tensor product, matrices
Article copyright: © Copyright 1971 American Mathematical Society