$D+M$ constructions with general overrings.

Brewer, J. W.; Rutter, Edgar

doi:10.1307/mmj/1029001619

Cited by 125 publications

(62 citation statements)

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“…It was natural at this stage of knowledge to investigate the behaviour of the star operations in a general pullback setting and with respect to surjective homomorphisms of integral domains, after various different results concerning distinguished star operations (like the v-, the t-, or the w-operation) and particular "composite-type" constructions were obtained by different authors (cf., for instance, [3][4][5]7,11,12,15,17,20,24,33,38,42], and the survey papers [10,25]). …”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Star operations and pullbacks

Fontana

Park

2004

Journal of Algebra

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In this paper we study the star operations on a pullback of integral domains. In particular, we\ud characterize the star operations of a domain arising froma pullback of “a general type” by introducing\ud new techniques for “projecting” and “lifting” star operations under surjective homomorphisms of\ud integral domains. We study the transfer in a pullback (or with respect to a surjective homomorphism)\ud of some relevant classes or distinguished properties of star operations such as v−, t−, w−, b−,\ud d−, finite type, e.a.b., stable, and spectral operations. We apply part of the theory developed here to\ud give a complete positive answer to a problem posed by D.F. Anderson in 1992 concerning the star\ud operations on the “D +M” constructions

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Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

Star operations and pullbacks

Fontana

Park

2004

Journal of Algebra

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“…Partial answers regarding when a pullback ring is coherent were provided only for special cases of domains (see [11,18,21]). …”

Section: ) R Is An Arithmetical Ring If and Only If T Is An Arithmetimentioning

confidence: 99%

“…Then T = K + M, where M is a maximal ideal of T . Let D be a subring of K and set R = D + M. This general construction and the particular case where T = K [x] = K + x K [x], with K a field, D a subring of K such that Q(D) = K and R = D + x K [x], were investigated in[11]. Costa, Mott and Zafrullah[15] considered the case where R = D + x D S [x], where D is a ring and S is any multiplicatively closed subset of D. Before moving on to D + M constructions that admit zero divisors, we reproduce the following result, which classifies when this D + M construction is a Prüfer domain.…”

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confidence: 99%

Prüfer Conditions in Commutative Rings

Glaz

Schwarz

2011

Arab J Sci Eng

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This article explores several extensions of the Prüfer domain notion to rings with zero divisors. These extensions include semihereditary rings, rings with weak global dimension less than or equal to 1, arithmetical rings, Gaussian rings, locally Prüfer rings, strongly Prüfer rings, and Prüfer rings. The renewed interest in these properties, due to their connection to Kaplansky's Conjecture, has resulted in a large body of results shedding new light on the area. We survey the work done in this direction in the last 15 years, including results, examples and counterexamples, and a multitude of open problems.

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“…(4) S is a strong Mori semigroup if and only if each w-ideal of S is of finite type. (5) Let S be a strong Mori semigroup and let P ∈ w-max(S).…”

Section: Lemmamentioning

confidence: 99%

“…Put k = R, the field of real numbers.) Let φ : T → T /M be the natural projection and let R = φ −1 (k).Then R is neither Krull (since it is not integrally closed) nor Noetherian (cf [4,. Theorem 4] and[10, Theorem 2.3]), but R is an SM domain by[21, Proposition 3.7].…”

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confidence: 99%

Group rings and semigroup rings over strong Mori domains, II

Park

2004

Journal of Algebra

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Let R be an integral domain and let S be a torsion-free cancellative additive monoid with quotient group G. We show that the semigroup ring R[X; S] is a strong Mori domain if and only if R is a strong Mori domain, S is a strong Mori semigroup, and each nonzero element of G is of type (0, 0, 0, . . .).  2004 Elsevier Inc. All rights reserved.For an associative ring R and a semigroup S (written additively), N. Jacobson [18, p. 95] defines the semigroup ring of S over R to be the set of functions f from S into R that are finitely nonzero, where addition and multiplication are defined by the rules(s) = t +u=s f (t)g(u). Following D.G. Northcott [20, p. 128], we denote the semigroup ring of S over R by the symbol R[X; S]; we write the elements of R[X; S] in the form f i X s i , f i ∈ R, s i ∈ S. A general reference on semigroups and semigroup rings is [12]. As we are dealing only with semigroup rings R[X; S] that are integral domains, we impose the corresponding restrictions on R and S (cf. [12, Theorem 8.1]). Namely, we assume that R is an integral domain with quotient field K, S a torsion-free cancellative additive semigroup with 0 (i.e., monoid) and G the quotient group of S.

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$D+M$ constructions with general overrings.

Cited by 125 publications

References 0 publications

Star operations and pullbacks

Star operations and pullbacks

Prüfer Conditions in Commutative Rings

Group rings and semigroup rings over strong Mori domains, II

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