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Idempotent Ideals and Noetherian Polynomial Rings

Published online by Cambridge University Press:  20 November 2018

Charles Lanski
Charles Lanski*
Affiliation:
Department of Mathematics University of Southern California, Los Angeles, CA 90007
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Abstract

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If R is a commutative Noetherian ring and I is a nonzero ideal of R, it is known that R+I[x] is a Noetherian ring exactly when I is idempotent, and so, when R is a domain, I = R and R has identity. In this paper, the noncommutative analogues of these results, and the corresponding ones for power series rings, are proved. In the general case, the ideal I must satisfy the idempotent condition that TI = T for each right ideal T of R contained in I. It is also shown that when every ideal of R satisfies this condition, and when R satisfies the descending chain condition on right annihilators, R must be a finite direct sum of simple rings with identity.

Keywords

16A3316A3416A1216A48
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 1 , 01 March 1982 , pp. 48 - 53
Copyright
Copyright © Canadian Mathematical Society 1982

References

1. Gilmer, R. W., If R[x] is Noetherian, R contains an identity, Amer. Math. Monthly, 74 (1967), p. 700.Google Scholar
2. Gilmer, R. W., An existence theorem for non-Noetherian rings, Amer. Math. Monthly, 77 (1970), 621-623.Google Scholar
3. Robson, J. C., 'Simple Noetherian rings need not have unity elements, Bull. London Math. Soc, 7 (1975), 269-270.Google Scholar