The polynomial ring over a Goldie ring need not be a Goldie ring. (English) Zbl 0719.16015
A ring is right Goldie if it satisfies the ascending chain condition on right annihilator ideals and contains no infinite direct sum of right ideals. This paper is concerned with whether a polynomial ring over a right Goldie ring need be right Goldie. A commutative ring R is constructed such that R is Goldie but the polynomial ring R[x] is not.
Reviewer: A.K.Boyle (Washington)
MSC:
| 16P60 | Chain conditions on annihilators and summands: Goldie-type conditions |
| 16S36 | Ordinary and skew polynomial rings and semigroup rings |
| 13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
References:
| [1] | Camillo, V.; Guralnick, R., Polynomial rings over Goldie rings are often Goldie, (Proc. Amer. Math. Soc., 98 (1986)), 567-568, No. 4 · Zbl 0608.16020 |
| [2] | Kerr, J. Wald, An Example of a Goldie ring whose matrix ring is not Goldie, J. Algebra, 61, 590-592 (1979) · Zbl 0441.16014 |
| [3] | Shock, R. C., Polynomial rings over finite dimensional rings, Pacific J. Math., 42, 251-257 (1972) · Zbl 0213.04301 |
| [4] | Small, L. W., Orders in Artinian rings, J. Algebra, 4, 13-41 (1966) · Zbl 0147.28703 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
