Geodesic
Idempotent Ideals and Noetherian Polynomial Rings
Canadian mathematical bulletin, Tome 25 (1982) no. 1, pp. 48-53

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DOI

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.
DOI : 10.4153/CMB-1982-006-5
Mots-clés : 16A33, 16A34, 16A12, 16A48 
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
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     title = {Idempotent {Ideals} and {Noetherian} {Polynomial} {Rings}},
     journal = {Canadian mathematical bulletin},
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     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1982-006-5/}
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