Journal article
Almost Splitting Sets and AGCD Domains
Communications in Algebra, Vol.32(1), pp.147-158
2004
DOI: 10.1081/agb-120027857
Abstract
Let D be an integral domain. A multiplicative set S of D is an almost splitting set if for each 0 ≠ d ∈ D, there exists an n = n(d) with dn = st where s ∈ S and t is v-coprime to each element of S. An integral domain D is an almost GCD (AGCD) domain if for every x, y ∈ D, there exists a positive integer n = n(x, y) such that xnD ∩ ynD is a principal ideal. We prove that the polynomial ring D[X] is an AGCD domain if and only if D is an AGCD domain and D[X] ⊆ D′[X] is a root extension, where D′ is the integral closure of D. We also show that D + XDS[X] is an AGCD domain if and only if D and D S[X] are AGCD domains and S is an almost splitting set.
Details
- Title: Subtitle
- Almost Splitting Sets and AGCD Domains
- Creators
- D.D. AndersenT. Dumitrescu - University of BucharestM. Zafrullah - Idaho State University
- Resource Type
- Journal article
- Publication Details
- Communications in Algebra, Vol.32(1), pp.147-158
- Publisher
- Marcel Dekker Inc.
- DOI
- 10.1081/agb-120027857
- ISSN
- 0092-7872
- Comment
- References: Andersen, D.D., Zafrullah, M., Almost Bézout domains (1991) J. Algebra, 142, pp. 285-309; Anderson, D.D., Zafrullah, M., Splitting sets in integral domains (2001) Proc. Amer. Math. Soc., 129, pp. 2209-2217; Anderson, D.D., Anderson, D.F., Zafrullah, M., Splitting the t-class group (1991) J. Pure Appl. Algebra, 74, pp. 17-37; Anderson, D.D., Anderson, D.F., Zafrullah, M., Rings between D[X] and K[X] (1991) Houston J. Math., 17, pp. 109-129; Anderson, D.D., Anderson, D.F., Zafrullah, M., Atomic domains in which almost all atoms are prime (1992) Comm. Algebra, 20, pp. 1447-1462; Anderson, D.D., Mott, J.L., Zafrullah, M., Finite character representations for integral domains (1992) Boll. Un. Mat. Ital. B (7), 6, pp. 613-630; Anderson, D.D., Anderson, D.F., Zafrullah, M., The ring D + XDS[X] and t-splitting sets (2001) Arab. J. Sci. Eng. Sect. C Theme Issues, 26, pp. 3-16; Costa, D., Mott, J.L., Zafrullah, M., The construction D + XDS[X] (1978) J. Algebra, 53, pp. 423-439; Dumitrescu, T., Zafrullah, M., LCM-splitting sets in some ring extensions (2002) Proc. Amer. Math. Soc., 130, pp. 1639-1644; Dumitrescu, T., Lequain, Y., Mott, J.L., Zafrullah, M., Almost GCD domains of finite t-character (2001) J. Algebra, 245, pp. 161-181; Hedstrom, J., Houston, E., Some remarks on star-operations (1980) J. Pure Appl. Algebra, 18, pp. 37-44; Mott, J.L., Convex directed subgroups of a group of divisibility (1974) Canad. J. Math., 26, pp. 532-542; Mott, J.L., Schexnayder, M., Exact sequences of semi-value groups (1976) J. Reine Angew. Math., 283-284, pp. 388-401; Uda, H., LCM-stableness in ring extensions (1983) Hiroshima Math. J., 13, pp. 357-377; Zafrullah, M., A general theory of almost factoriality (1985) Manuscripta Math., 51, pp. 29-62; Zafrullah, M., The D + XDS[X] construction from GCD- domains (1988) J. Pure Appl. Algebra, 50, pp. 93-107
- Language
- English
- Date published
- 2004
- Academic Unit
- Mathematics
- Record Identifier
- 9984230626302771
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