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Nil Semi-Groups of Rings with a Polynomial Identity

Published online by Cambridge University Press:  22 January 2016

S. A. Amitsur
S. A. Amitsur*
Affiliation:
The Hebrew University and The University of Chicago
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The basic properties of associative rings R satisfying a polynomial identity p[x1…, xn] = 0 were obtained under the assumptions that the ring was an algebra [e.g., [4] Ch. X], or with rather strong restrictions on the ring of operators ([1]). But it is desirable to have these properties for arbitrary rings, and the present paper is the first of an attempt in this direction. The problem is almost trivial for prime or semi-prime rings but quite difficult in arbitrary rings. The known proofs for algebras have to be modified and in some cases new proofs have to be obtained as the existing proofs fail to exploit the known structure. In the present paper we extend the results of [1] on the nil subalgebras of a ring with an identity for arbitrary multiplicative nil semi-groups of the ring and for arbitrary rings.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 27 , Issue 1 , February 1966 , pp. 103 - 111
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1966

References

[1] Amitsur, S. A., Nil PI-rings Proc. Am. Math. Soc. 2 (1951), pp. 538–90.Google Scholar
[2] Amitsur, S. A., Rings with a pivotal monomial, Proc. Am. Math. Soc, 9 (1958). pp. 635–42.Google Scholar
[3] Drazin, M. P., A generalization of polynomial identities in rings, Proc. Am. Math. Soc, 8 (1957), pp. 352–61.Google Scholar
[4] Jacobson, N., Structure of Rings, Am. Math. Soc Colloquium Pub. No. 37, Providence, R. I., 1956. (Ch. X, p. 232).Google Scholar