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Quotient rings of polynomial rings

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Abstract

Let R be a commutative ring with identity. The multiplicatively closed sets U2={f∈R[X]: c(f)−1=R}, (U2)={f∈U2: f is regular} and S={f∈R[X]: c(f)=R} are studied. By considering various equalities between these sets, many characterizations of Noetherian rings are found. In particular, a Noetherian ring R has depth ≤1 if and only if S=(U2): and each maximal ideal of a Noetherian ring is regular if and only if U2=(U2).

The theory of Prüfer v-multiplication rings (PVMR's) is developed for rings with zero divisors. Six equivalent conditions are given to the statement that an additively regular v-ring R is a PVMR.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, 65211, Columbia, MO, USA

    James A. Huckaba & Ira J. Papick

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  1. James A. Huckaba
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  2. Ira J. Papick
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Huckaba, J.A., Papick, I.J. Quotient rings of polynomial rings. Manuscripta Math 31, 167–196 (1980). https://doi.org/10.1007/BF01303273

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  • DOI: https://doi.org/10.1007/BF01303273

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