Idempotent Ideals and Noetherian Polynomial Rings
Canadian mathematical bulletin, Tome 25 (1982) no. 1, pp. 48-53
Voir la notice de l'article provenant de la source Cambridge University Press
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.
Lanski, Charles. Idempotent Ideals and Noetherian Polynomial Rings. Canadian mathematical bulletin, Tome 25 (1982) no. 1, pp. 48-53. doi: 10.4153/CMB-1982-006-5
@article{10_4153_CMB_1982_006_5,
author = {Lanski, Charles},
title = {Idempotent {Ideals} and {Noetherian} {Polynomial} {Rings}},
journal = {Canadian mathematical bulletin},
pages = {48--53},
year = {1982},
volume = {25},
number = {1},
doi = {10.4153/CMB-1982-006-5},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1982-006-5/}
}
[1] 1. Gilmer, R. W., If R[x] is Noetherian, R contains an identity, Amer. Math. Monthly, 74 (1967), p. 700. Google Scholar
[2] 2. Gilmer, R. W., An existence theorem for non-Noetherian rings, Amer. Math. Monthly, 77 (1970), 621-623. Google Scholar
[3] 3. Robson, J. C., 'Simple Noetherian rings need not have unity elements, Bull. London Math. Soc, 7 (1975), 269-270. Google Scholar
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