Primary decomposition of divisorial ideals in Mori domains

Houston, Evan; Lucas, Thomas G.; Viswanathan, T.

doi:10.1016/0021-8693(88)90110-x

Cited by 29 publications

(6 citation statements)

References 4 publications

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“…(6) It is clear that P + S P is the maximal proper prime ideal of S P . Since S is a Mori semigroup and P is divisorial (by (5)), the proof of (6) is similar to [17,Proposition 1.1].…”

Section: Proofmentioning

confidence: 88%

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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“…(6) It is clear that P + S P is the maximal proper prime ideal of S P . Since S is a Mori semigroup and P is divisorial (by (5)), the proof of (6) is similar to [17,Proposition 1.1].…”

Section: Proofmentioning

confidence: 88%

Group rings and semigroup rings over strong Mori domains, II

Park

2004

Journal of Algebra

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“…Proposition 2.11. Hence if A is not Archimedean, then there is a n element .r E A n U(A*) \ U(A), and for any prime ideal P of A with [HLV,Theorem 2.11. We claim that (a,b)B # B. Granting the claim, let Q be a prime of B with a, b E Q.…”

Section: Lying Over In Almost Integral Extensionsmentioning

confidence: 97%

On the prime spectrum of a mori domain

Barucci

Houston

1996

Communications in Algebra

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“…Suppose that we take l = 1 in Family 1 or 3, so M 2 is universal. For n < 100, this gives local CK domains with M 2 universal for n = 1, 3,4,5,6,7,8,9,10,12,13,14,15,17,18,20,21,24,26,28,30,31,32,33,38,40,42,44,48 [10] and Clark, Govasi, and Pollack [7] gave some examples in characteristic 0 from number theory. We begin with this following example.…”

Section: Familymentioning

confidence: 99%

Atoms in Quasilocal Integral Domains

Anderson¹,

Bombardier²

2018

Preprint

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Let (R, M) be a quasilocal integral domain. We investigate the set of irreducible elements (atoms) of R. Special attention is given to the set of atoms in M\M 2 and to the existence of atoms in M 2 . While our main interest is in local Cohen-Kaplansky (CK) domains (atomic integral domains with only finitely many nonassociate atoms), we endeavor to obtain results in the greatest generality possible. In contradiction to a statement of Cohen and Kaplansky, we construct a local CK domain with precisely eight nonassociate atoms having an atom in M 2 .

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Primary decomposition of divisorial ideals in Mori domains

Cited by 29 publications

References 4 publications

Group rings and semigroup rings over strong Mori domains, II

Group rings and semigroup rings over strong Mori domains, II

On the prime spectrum of a mori domain

Atoms in Quasilocal Integral Domains

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