The \(D+XD_ s[X]\) construction from GCD-domains. (English) Zbl 0656.13020

Let A be a domain, S a multiplicative subset of A and X an indeterminate over A. In this paper the author studies the ring \(A^{(S)}=\{a_ 0+\sum_{i\geq 1}a_ iX^ i| \quad a_ 0\in A,\quad a_ i\in S^{-1}A\}\) assuming that A is a GCD-domain. In particular, he shows that the behaviour of \(A^{(S)}\) depends on the relationship between S and the prime ideals P of A such that \(A_ P\) is a valuation ring. As an application he constructs examples of locally GCD-domains which are not GCD-domains.
Reviewer: L.Bădescu


13F15 Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial)
13B02 Extension theory of commutative rings
13G05 Integral domains
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