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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Tensor products of polynomial identity algebras
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by Elizabeth Berman PDF
Trans. Amer. Math. Soc. 156 (1971), 259-271 Request permission


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.
  • I. N. Herstein, Noncommutative rings, The Carus Mathematical Monographs, No. 15, Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968. MR 0227205
  • Nathan Jacobson, Structure of rings, Revised edition, American Mathematical Society Colloquium Publications, Vol. 37, American Mathematical Society, Providence, R.I., 1964. MR 0222106
  • Claudio Procesi and Lance Small, Endomorphism rings of modules over $\textrm {PI}$-algebras, Math. Z. 106 (1968), 178–180. MR 233846, DOI 10.1007/BF01110128
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 156 (1971), 259-271
  • MSC: Primary 16.49
  • DOI:
  • MathSciNet review: 0274515