Prüfer ⋆–multiplication domains and ⋆–coherence

Fontana, Marco; Picozza, Giampaolo

doi:10.1007/s11587-006-0010-1

Cited by 10 publications

(4 citation statements)

References 31 publications

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“…This concept extends that of Prüfer * -multiplication domain (P * MD), for a given semistar operation * on D, [5]. By the above, a domain D is a (M, )-multiplication domain if and only if it is a ( f ) D -domain (i.e., a domain in which each finitely generated ideal is ( f ) D -invertible, see [11]). When =˜ we can say something more (cf.…”

Section: Proposition 54 Let D Be An Integral Domain Which Is Not Amentioning

confidence: 89%

Star operations on modules

Baghdadi

Picozza

2009

Journal of Algebra

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We present a theory of (semi)star operations for torsion-free modules. This extends the analogous theory of star operations on domains as in [R. Gilmer, Multiplicative Ideal Theory, M. Dekker, New York, 1972] and its generalization to semistar operations studied in [A. Okabe, R. Matsuda, Semistar operations on integral domains, Math. J. Toyama Univ. 17 (1994) 1-21], and recovers some closure operations defined on modules. We investigate some properties of (semi)star operations on a given module over a domain D and their relation with the properties of some classes of semistar operations induced on D. Among other things, this leads to a connection between semistar operations on the D-module and localizing systems on the domain D.

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Section: Proposition 54 Let D Be An Integral Domain Which Is Not Amentioning

confidence: 89%

Star operations on modules

Baghdadi

Picozza

2009

Journal of Algebra

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“…The equivalences (i)⇔(ii)⇔(iii) follow from [29,Proposition 11]. (iii)⇒(iv) Since a * -ideal is trivially also a * f -ideal, thus under the assumption (iii), Inv * (D) ⊆ Inv * f (D).…”

Section: V-class Groups and Valuation Domainsmentioning

confidence: 91%

Some remarks on Prüfer ⋆-multiplication domains and class groups

Anderson¹,

Fontana²,

Zafrullah³

2008

Journal of Algebra

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Let D be an integral domain with quotient field K and let X be an indeterminate over D. Also, let T T T := {T λ | λ ∈ Λ} be a defining family of quotient rings of D and suppose that * is a finite type star operation on D induced by T T T . We show that D is a P * MD (respectively, PvMD) if and only if (c D (f g)) * = (c D (f )c D (g)) * (respectively, (c D (f g)) w = (c D (f )c D (g)) w ) for all 0 = f, g ∈ K[X]. A more general version of this result is given in the semistar operation setting. We give a method for recognizing PvMD's which are not P * MD's for a certain finite type star operation * . We study domains D for which the * -class group Cl * (D) equals the t-class group Cl t (D) for any finite type star operation * , and we indicate examples of PvMD's D such that Cl * (D) Cl t (D). We also compute Cl v (D) for certain valuation domains D.

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“…For example, Zafrullah ([23, Theorem 8]) proved that an integral domain R is a Prüfer domain if and only if R is an integrally closed domain such that the t-operation and d-operation coincide, and Wang and McCasland ( [16,Theorem 5.4]) proved that R is a Krull domain if and only if R is a completely integrally closed domain such that the w-operation and v-operation coincide. In a similar way, semistar operations have received significant interest in the field of multiplicative ideal theory (for instance, see [4], [6], [7], [8], [17], [19], [20], [21], to name only a few).…”

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confidence: 99%

Descending chains of semistar operations

Choi¹,

McEldowney²,

Walker³

2017

Preprint

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A class of integer-valued functions defined on the set of ideals of an integral domain R is investigated. We show that this class of functions, which we call ideal valuations, are in one-to-one correspondence with countable descending chains of finite type, stable semistar operations with largest element equal to the e-operation. We use this class of functions to recover familiar semistar operations such as the w-operation and to give a solution to a conjecture by Chapman and Glaz when the ring is a valuation domain.1991 AMS Mathematics subject classification. 13A15. Keywords and phrases. semistar operations, localizing systems, valuation domains.

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Prüfer ⋆–multiplication domains and ⋆–coherence

Cited by 10 publications

References 31 publications

Star operations on modules

Star operations on modules

Some remarks on Prüfer ⋆-multiplication domains and class groups

Descending chains of semistar operations

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