## 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

### MSC:

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

### Keywords:

valuation rings; GCD-domain
Full Text:

### References:

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