Prime ideals in polynomial rings over one-dimensional domains

Heinzer, William; Wiegand, Sylvia

doi:10.1090/S0002-9947-1995-1242087-5

Prime ideals in polynomial rings over one-dimensional domains
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by William Heinzer and Sylvia Wiegand PDF

Trans. Amer. Math. Soc. 347 (1995), 639-650 Request permission

Abstract:

Let $R$ be a one-dimensional integral domain with only finitely many maximal ideals and let $x$ be an indeterminate over $R$. We study the prime spectrum of the polynomial ring $R[x]$ as a partially ordered set. In the case where $R$ is countable we classify $\operatorname {Spec} (R[x])$ in terms of splitting properties of the maximal ideals ${\mathbf {m}}$ of $R$ and the valuative dimension of ${R_{\mathbf {m}}}_{}$.

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