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The ideal of polynomials vanishing on a commutative ring

Author(s): Robert Gilmer
Journal: Proc. Amer. Math. Soc. 127 (1999), 1265-1267.
MSC (1991): Primary 13B25; Secondary 13E10
Posted: January 27, 1999
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Abstract: We determine equivalent conditions on a commutative Artinian ring $S$ in order that the ideal of $S[t]$ consisting of polynomials that vanish on $S$ should be principal. Our results correct an error in a paper of Niven and Warren.

References:

[J]
G. Jacob, Anneau de fonctions polynomes d'un anneau commutatif unitaire, Commun. Algebra 8 (1990), 793-811. MR 82j:13007

[N]
W. Narkiewicz, Polynomial Mappings, Lecture Notes in Math. 1600 (1995). MR 97e:11037

[NW]
I. Niven and D. Warren, A generalization of Fermat's Theorem, Proc. Amer. Math. Soc. 8 (1957), 306-313.

[S]
E. Snapper, Completely primary rings I., Annals of Math. 52 (1950), 666-693. MR 12:314b

[ZS]
O. Zariski and P. Samuel, Commutative Algebra, vol. I, Springer, Berlin-Heidelberg, 1986.

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Additional Information:

Robert Gilmer
Affiliation: Department of Mathematics, Florida State University, Tallahassee, Florida 32306-4510
Email: gilmer@math.fsu.edu

DOI: 10.1090/S0002-9939-99-04634-1
PII: S 0002-9939(99)04634-1
Keywords: Vanishing polynomials, Artinian rings
Received by editor(s): June 10, 1997
Received by editor(s) in revised form: August 6, 1997
Posted: January 27, 1999
Communicated by: Wolmer V. Vasconcelos
Copyright of article: Copyright 1999, American Mathematical Society

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