Star operations and pullbacks

Fontana, Marco; Park, Mi Hee

doi:10.1016/j.jalgebra.2003.10.002

Cited by 12 publications

(21 citation statements)

References 35 publications

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“…The main goal of this work is to establish functorial relations among the star class groups of R, D, and T , by using the theory that we have recently developed in [22] concerning the "lifting" and the "projection" of a star operation under a surjective homomorphism of integral domains, the "extension" of a star operation to its overrings and the "glueing" of star operations in pullback diagrams of a rather general type. One of the principal results proven in this paper is that, given a pullback diagram of type ( + ) and a star operation * of finite type on R, if * ϕ denotes the "projection" of * onto D (respectively, ( * ) T denotes the "extension" of * to T ), under a mild hypothesis on the group of units of T , the sequence of canonical homomorphisms…”

Section: Introduction and Background Resultsmentioning

confidence: 99%

“…If 1 and 2 are two star operations on D with 1 2 [4, page 811]). However, there is a special submonoid of F (D) which reverses the inclusion: In [22] we considered the problem of "lifting a star operation" with respect to a surjective ring homomorphism between two integral domains. More precisely: …”

Section: Introduction and Background Resultsmentioning

confidence: 99%

“…In [22] we also considered the problem of "projecting a star operation" with respect to a surjective homomorphism of integral domains, with particular emphasis on pullback constructions of a "special" kind. More precisely: Lemma 1.5 [22, Propositions 2.6, 2.7, 2.9 and Theorem 2.12].…”

Section: Introduction and Background Resultsmentioning

confidence: 99%

“…For this purpose, recall that, in [22], we also considered the problem of "extending a star operation" defined on an integral domain R to some overring T of R. We need the following notation. Let * be a star operation on an integral domain R and let T be an overring of R such that (R : T ) = 0.…”

Section: Resultsmentioning

confidence: 99%

“…Set := ϕ ∧ ι . We know that is a star operation on R [22,Corollary 2.5]. If (( ϕ ) T ) f = (( ϕ ) f ) T (e.g., this hypothesis is satisfied in each one of the following cases: …”

Section: )] H (T R ) T = H T R ⊇ (H R(m)) T R(m) ∩ (H T ) T T = (I Tmentioning

confidence: 99%

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On the star class group of a pullback

Fontana

Park

2005

Journal of Algebra

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For the domain $R$ arising from the construction $T, M, D$,\ud we relate the star class groups of $R$ to those of $T$ and $D$.\ud More precisely, let $T$ be an integral domain, $M$ a nonzero maximal ideal of $T$, $D$ a proper subring of $k:=T/M$, $\varphi: T\rightarrow k$ the natural projection, and let $R={\varphi}^{-1}(D)$.\ud For each star operation $\ast$ on $R$, we define the star operation\ud $\ast_\varphi$ on $D$, i.e., the ``projection'' of $\ast$ under $\varphi$, and the star operation ${(\ast)}_{_{\!T}}$ on $T$, i.e., the ``extension'' of $\ast$ to $T$.\ud Then we show that, under a mild hypothesis on the group of units of\ud $T$, if $\ast$ is a star operation of finite type, then the sequence of canonical homomorphisms\ud $0\rightarrow \Cl^{\ast_{\varphi}}(D) \rightarrow \Cl^\ast(R)\ud \rightarrow \Cl^{{(\ast)}_{_{\!T}}}(T)\rightarrow 0$\ud is split exact. In particular, when $\ast = t_{R}$, we deduce\ud that the sequence\ud $\ud 0\rightarrow \Cl^{t_{D}}(D)\ud {\rightarrow} \Cl^{t_{R}}(R)\ud {\rightarrow}\Cl^{(t_{R})_{_{\!T}}}(T) \rightarrow 0\ud $\ud is split exact. The relation between ${(t_{R})_{_{\!T}}}$ and\ud $t_{T}$ (and between $\Cl^{(t_{R})_{_{\!T}}}(T)$ and $\Cl^{t_{T}}(T)$)\ud is also investigated

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Section: Introduction and Background Resultsmentioning

confidence: 99%

Section: Introduction and Background Resultsmentioning

confidence: 99%

Section: Introduction and Background Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

“…Set := ϕ ∧ ι . We know that is a star operation on R [22,Corollary 2.5]. If (( ϕ ) T ) f = (( ϕ ) f ) T (e.g., this hypothesis is satisfied in each one of the following cases: …”

Section: )] H (T R ) T = H T R ⊇ (H R(m)) T R(m) ∩ (H T ) T T = (I Tmentioning

confidence: 99%

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On the star class group of a pullback

Fontana

Park

2005

Journal of Algebra

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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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Multiplicative closure operations on ring extensions

Spirito

2021

Journal of Pure and Applied Algebra

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Star operations and pullbacks

Cited by 12 publications

References 35 publications

On the star class group of a pullback

On the star class group of a pullback

Some remarks on Prüfer ⋆-multiplication domains and class groups

Multiplicative closure operations on ring extensions

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