Tensor products of polynomial identity algebras
Author:
Elizabeth Berman
Journal:
Trans. Amer. Math. Soc. 156 (1971), 259271
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 in A, . We call A a Palgebra 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, , the set of nbyn 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, satisfies some power of a standard identity. We generalize this result to Palgebras over commutative rings with identity. It follows that if A is a Palgebra, A satisfies a power of the docile identity.
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Claudio
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178–180. MR 0233846
(38 #2167)
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 I. N. Herstein, Noncommutative rings, Carus Math. Monographs, no. 151, Math. Assoc. of America; distributed by Wiley, New York, 1968. MR 37 #2790. MR 0227205 (37:2790)
 [2]
 Nathan Jacobson, Structure of rings, 2nd 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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 Claudio Procesi and Lance Small, Endomorphism rings of modules over PIalgebras, Math. 106 (1968), 178180. MR 38 #2167. MR 0233846 (38:2167)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719710274515X
PII:
S 00029947(1971)0274515X
Keywords:
Polynomial identity,
tensor product,
matrices
Article copyright:
© Copyright 1971
American Mathematical Society
