Krull rings with zero divisors

Portelli, Dario; Spangher, Walter

doi:10.1080/00927878308822934

Cited by 18 publications

(6 citation statements)

References 9 publications

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“…For the definition of a * -operation, the reader may consult [4]. There is a considerable amount of research devoted to extending multiplicative ideal theory to commutative rings containing zero divisors, see for example [7,8,10,11,12,15]. Recently, the subject of the w-operation has generated considerable interest.…”

Section: Is Called a W-module If M W = M And M Is Said To Be A W-mentioning

confidence: 99%

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Ω-Modules OVER COMMUTATIVE RINGS

Yin¹,

Wang²,

Zhu³

et al. 2011

Journal of the Korean Mathematical Society

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Abstract. Let R be a commutative ring and let M be a GV -torsionfree R-module. Then M is said to be a w-module if Ext 1 R (R/J, M ) = 0 for any J ∈ GV (R), and the w-envelope of M is defined by Mw = {x ∈ E(M ) | Jx ⊆ M for some J ∈ GV (R)}. In this paper, w-modules over commutative rings are considered, and the theory of w-operations is developed for arbitrary commutative rings. As applications, we give some characterizations of w-Noetherian rings and Krull rings. IntroductionLet R be a domain with quotient field K, and let F (R) be the set of nonzero fractional ideals of R.Recall from [17] that for a domain R and a torsionfree R-module M , the w-envelope of M is defined by is a * -operation called the w-operation.One can see that the notion of a w-ideal coincides with the notion of a semi-divisorial ideal introduced by Glaz and Vasconcelos in 1977 [5] which may have some far reaching effects on the theory of * -operations. As a * -operation, the w-operation was briefly yet effectively touched on by Hedstrom and Houston in 1980 under the name of F ∞ -operation [6]. Later, this * -operation was intensely studied by Wang and McCasland in a more general setting. In particular, Wang and McCasland showed that the w-envelope notion is a very useful tool in studying strong Mori domains [17,18]. For the definition of a * -operation, the reader may consult [4]. There is a considerable amount of research devoted to extending multiplicative ideal theory to commutative rings containing zero divisors, see for example

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Section: Is Called a W-module If M W = M And M Is Said To Be A W-mentioning

confidence: 99%

“…The inverse can be proved similarly. □ Before moving to another topic, we should note that different definitions of * -operation on arbitrary commutative rings appeared in the literatures [7], [8] and [15], but our "w-operation" satisfies all of them.…”

Section: ) It Follows From the Fact Thatmentioning

confidence: 99%

Ω-Modules OVER COMMUTATIVE RINGS

Yin¹,

Wang²,

Zhu³

et al. 2011

Journal of the Korean Mathematical Society

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“…For more information on Mori rings we refer to the work of Lucas [29,30,31], and in particular to the characterization given in [29,Theorem 2.22]. For a characterization of Krull rings we refer to the work of Kang [27,Theorem 13] (note that older concepts of Krull rings -as given in [34] and in Huckaba's monograph [24] -coincide with the present one in the setting of Marot rings).…”

Section: Preliminaries On Marot Mori and Krull Ringsmentioning

confidence: 99%

“…(1) The fact that R 1 × R 2 is a Marot ring if and only if both R 1 and R 2 are Marot rings is easy and well-known (see [34,Proposition 4]). We verify the statement for v-Marot rings.…”

Section: Proposition 42 (Arithmetic Properties)mentioning

confidence: 99%

“…Krull rings with zero divisors were introduced independently by Kennedy [28] and by Portelli and Spangher [34], and they found a solid treatment in Huckaba's monograph [24] in the setting of Marot rings. The theory of Krull and Mori rings was further developed (even without requiring the Marot property) by Chang, Glaz, Kang, Lucas, and others ( [7,8,9,19,25,26,27,29,30,31,33]).…”

Section: Introductionmentioning

confidence: 99%

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ON v-MAROT MORI RINGS AND C-RINGS

Geroldinger¹,

Ramacher²,

Reinhart³

2015

Journal of the Korean Mathematical Society

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Abstract. C-domains are defined via class semigroups, and every Cdomain is a Mori domain with nonzero conductor whose complete integral closure is a Krull domain with finite class group. In order to extend the concept of C-domains to rings with zero divisors, we study v-Marot rings as generalizations of ordinary Marot rings and investigate their theory of regular divisorial ideals. Based on this we establish a generalization of a result well-known for integral domains. Let R be a v-Marot Mori ring, R its complete integral closure, and suppose that the conductor f = (R : R) is regular. If the residue class ring R/f and the class group C( R) are both finite, then R is a C-ring. Moreover, we study both v-Marot rings and C-rings under various ring extensions.

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Die Klassengruppe einer kommutativen Ordnung

Halter‐Koch

1994

Mathematische Nachrichten

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Krull rings with zero divisors

Cited by 18 publications

References 9 publications

Ω-Modules OVER COMMUTATIVE RINGS

Ω-Modules OVER COMMUTATIVE RINGS

ON v-MAROT MORI RINGS AND C-RINGS

Die Klassengruppe einer kommutativen Ordnung

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