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
Full Text: DOI


[1] Brewer, J.; Heinzer, W., Associated primes of principal ideals, Duke math. J., 41, 1-7, (1974) · Zbl 0284.13001
[2] Cohn, P., Bezout rings and their subrings, Math. proc. Cambridge philos. soc., 64, 251-264, (1968) · Zbl 0157.08401
[3] Cohn, P., Free rings and their relations, (1971), Academic Press New York · Zbl 0232.16003
[4] Costa, D.; Mott, J.; Zafrullah, M., The construction D + XDs[X], J. algebra, 53, 423-439, (1978) · Zbl 0407.13003
[5] Fuchs, L., Riesz groups, Ann. scuola norm. sup. Pisa, 19, 1-34, (1965) · Zbl 0125.28703
[6] Gilmer, R., Multiplicative ideal theory, (1972), Dekker New York · Zbl 0248.13001
[7] Griffin, M., Some results on v-multiplication rings, Canad. J. math., 19, 710-722, (1967) · Zbl 0148.26701
[8] Heinzer, W., An essential integral domain with a non essential localization, Canad. J. math., 33, 2, 400-403, (1981) · Zbl 0411.13013
[9] Heinzer, W.; Ohm, J., An essential ring which is not a v-multiplication ring, Canad. J. math., 25, 856-861, (1973) · Zbl 0258.13001
[10] Jaffard, P., LES systèmes d’ideaux, (1960), Dunod Paris · Zbl 0101.27502
[11] Mott, J.; Schexnayder, M., Exact sequences of semi-value groups, J. reine angew. math., 283/284, 388-401, (1976) · Zbl 0347.13001
[12] Mott, J.; Zafrullah, M., On Prüfer v-multiplication domains, Manuscripta math., 35, 1-26, (1981) · Zbl 0477.13007
[13] Nishimura, T., On the v-ideals of an integral domain, Bull. Kyoto univ. ed. ser. B, 17, 47-50, (1961)
[14] Sheldon, P., Prime ideals in GCD-domains, Canad. J. math., 26, 98-107, (1974) · Zbl 0247.13009
[15] Zafrullah, M., On finite conductor domains, Manuscripta math., 24, 191-203, (1978) · Zbl 0383.13013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.