Vol. 44, No. 1, 1973

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Power series rings over Prüfer domains

Jimmy T. Arnold

Vol. 44 (1973), No. 1, 1–11
Abstract

Let R be a commutative ring with identity. R is said to have dimension n, written dimR = n, if there exists a chain P0 P1 Pn of n + 1 prime ideals of R, where Pn R, but no such chain of n + 2 prime ideals. Seidenberg has shown that if dimR = n and X is an indeterminate over R, then n + 1 dimR[X] 2n + 1. Moreover, he has shown that dimR[X] = n + 1 if R is a Prüfer domain. The author has shown that if V is a rank one nondiscrete valuation ring, then dimV [[X]] = . The principal result of this paper is that if D is a Prüfer domain with dimD = n, then either dimD[[X]] = n + 1 or dimD[[X]] = , and necessary and sufficient conditions are given.

Mathematical Subject Classification 2000
Primary: 13J05
Milestones
Received: 20 September 1971
Published: 1 January 1973
Authors
Jimmy T. Arnold