*-NOETHERIAN DOMAINS AND THE RING D[X]<sub>N*</sub>, II

Chang, Gyu-Whan

doi:10.4134/jkms.2011.48.1.049

Cited by 3 publications

(3 citation statements)

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“…It is clear that the spectral map g a : Spec(R(T )) → Spec(R) is surjective. For uses of Nagata rings and related rings of rational functions in the context of star and semistar operations, see [28], [24], [25], [4], [5], [6], [7], [12], [34], [38], [37] and [45]. Now, we consider another map γ : Spec(R) → Spec(R(T )) by setting γ(P ) := P R(T ) for each P ∈ Spec(R): this map is well-defined and injective (since IR(T ) ∩ R = I, for all ideals I of R [28, Proposition 33.1(4)]).…”

Section: The Space Of Semigroup Primes Of the Nagata Ringmentioning

confidence: 99%

Topological properties of semigroup primes of a commutative ring

2017

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A semigroup prime of a commutative ring R is a prime ideal of the semigroup (R, Â·). One of the purposes of this paper is to study, from a topological point of view, the space S(R) of prime semigroups of R. We show that, under a natural topology introduced by B. Olberding in 2010, S(R) is a spectral space (after Hochster), spectral extension of Spec(R) , and that the assignment Râ\u86¦ S(R) induces a contravariant functor. We then relateâ\u80\u94in the case R is an integral domainâ\u80\u94the topology on S(R) with the Zariski topology on the set of overrings of R. Furthermore, we investigate the relationship between S(R) and the space X(R) consisting of all nonempty inverse-closed subspaces of Spec(R) , which has been introduced and studied in Finocchiaro et al. (submitted). In this context, we show that S(R) is a spectral retract of X(R) and we characterize when S(R) is canonically homeomorphic to X(R) , both in general and when Spec(R) is a Noetherian space. In particular, we obtain that, when R is a BÃ©zout domain, S(R) is canonically homeomorphic both to X(R) and to the space Overr(R) of the overrings of R (endowed with the Zariski topology). Finally, we compare the space X(R) with the space S(R(T)) of semigroup primes of the Nagata ring R(T), providing a canonical spectral embedding X(R) â\u86ª S(R(T)) which makes X(R) a spectral retract of S(R(T))

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Section: The Space Of Semigroup Primes Of the Nagata Ringmentioning

confidence: 99%

Topological properties of semigroup primes of a commutative ring

2017

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“…Clearly, each t-ideal is a * f -ideal, and thus each maximal * f -ideal is a t-ideal if and only if * w = w. For more on basic properties of star operations, see [3], [11], or [13, Sections 32 and 34]. It is known that a * f -quasi-Prüfer domain is a UMT-domain (Lemma 4((1) ⇒ (5)).…”

Section: Introductionmentioning

confidence: 99%

Kronecker Function Rings and Prüfer-Like Domains

Chang¹

2012

Korean Journal of Mathematics

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Let D be an integral domain,D be the integral closure of D, * be a star operation of finite character on D, * w be the so-called * w-operation on D induced by * , X be an indeterminate over D, N * = {f ∈ D[X]|c(f) * = D}, and Kr(D, *) = {0} ∪ { f g |0 = f, g ∈ D[X] and there is an 0 = h ∈ D[X] such that (c(f)c(h)) * ⊆ (c(g)c(h)) * }. In this paper, we show that D is a *-quasi-Prüfer domain if and only ifD[X] N * = Kr(D, * w). As a corollary, we recover Fontana-Jara-Santos's result that D is a Prüfer *-multiplication domain if and only if D[X] N * = Kr(D, * w).

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“…In [C,Section 3], the * -global transform of D is defined to be the set D * g = {x ∈ K | M 1 · · · M n x ⊆ D for some M i ∈ * s -Max(D)}. Then D * g is an overring of D and D * g = D * s g = D * w g , so the concept of * w -global transform coincides with that of * -global transform.…”

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confidence: 99%

A note on *_{𝑤}-Noetherian domains

Hwang

Lim

2012

Proc. Amer. Math. Soc.

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Let D be an integral domain with quotient field K, * be a staroperation on D, and GV * (D) be the set of finitely generated ideals J of D such that J * = D. Then the map * w defined by I * w = {x ∈ K | Jx ⊆ I for some J ∈ GV * (D)} for all nonzero fractional ideals I of D is a finite character staroperation on D. In this paper, we study several properties of * w -Noetherian domains. In particular, we prove the Hilbert basis theorem for * w -Noetherian domains.

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-NOETHERIAN DOMAINS AND THE RING D[X]_N, II

Abstract: Abstract. Let D be an integral domain with quotient field K, X be a nonempty set of indeterminates over D, * be a star operation on D, N * = {f ∈ D [X]

Cited by 3 publications

References 17 publications

Topological properties of semigroup primes of a commutative ring

Topological properties of semigroup primes of a commutative ring

Kronecker Function Rings and Prüfer-Like Domains

A note on *_{𝑤}-Noetherian domains

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*-NOETHERIAN DOMAINS AND THE RING D[X]N*, II

Abstract: Abstract. Let D be an integral domain with quotient field K, X be a nonempty set of indeterminates over D, * be a star operation on D, N * = {f ∈ D [X]

Cited by 3 publications

References 17 publications

Topological properties of semigroup primes of a commutative ring

Topological properties of semigroup primes of a commutative ring

Kronecker Function Rings and Prüfer-Like Domains

A note on *_{𝑤}-Noetherian domains

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Product

Resources

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-NOETHERIAN DOMAINS AND THE RING D[X]_N, II