Some finiteness conditions on the set of overrings of an integral domain

Gilmer, Robert

doi:10.1090/s0002-9939-02-06816-8

Cited by 54 publications

(18 citation statements)

References 9 publications

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“…(see [28] [1,2,3,4,5,6,7,8,9,13,14,15,16,17,21,20,22,25,26,29] [24]. A one-dimensional domain is Gorenstein if and only if the inverse of the maximal ideal is generated by two elements [10].…”

Section: Introductionmentioning

confidence: 99%

“…, and k[X p , X q , X r ] where p, q , and r are positive integers are extensively used as sources of examples and counter-examples in studying different properties of integral domains (see, for instance, [1,2,3,4,5,6,7,8,9,13,14,15,16,17,21,20,22,25,26,29]). The objective of this note is to study the divisoriality of those domains and present it as a unified reference for interested authors.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Note on the divisoriality of domains of the form $k[[X^{p}, X^{q}]]$, $k[X^{p}, X^{q}]$, $k[[X^{p}, X^{q}, X^{r}]]$, and $k[X^{p}, X^{q}, X^{r}]$

Mimouni¹

2016

Turk J Math

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Note on the divisoriality of domains of the form $k[[X^{p}, X^{q}]]$, $k[X^{p}, X^{q}]$, $k[[X^{p}, X^{q}, X^{r}]]$, and $k[X^{p}, X^{q}, X^{r}]$

Mimouni¹

2016

Turk J Math

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“…Proof. If T is integral over R, then by [20,Theorem 2.8], P := (R : T ) satisfies exactly one of the following:…”

Section: Conductor Of Integrally Closed Maximal Subringmentioning

confidence: 99%

Conch Maximal Subrings

Azarang

2020

Preprint

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It is shown that if R is a ring, p a prime element of an integral domain D ≤ R with ∞ n=1 p n D = 0 and p ∈ U (R), then R has a conch maximal subring (see [17]). We prove that either a ring R has a conch maximal subring or U (S) = S ∩ U (R) for each subring S of R (i.e., each subring of R is closed with respect to taking inverse, see [30]). In particular, either R has a conch maximal subring or U (R) is integral over the prime subring of R. We observe that if R is an integral domain with |R| = 2 2 ℵ 0 , then either R has a maximal subring or |M ax(R)| = 2 ℵ 0 , and in particular if in addition dim(R) = 1, then R has a maximal subring. If R ⊆ T be an integral ring extension, Q ∈ Spec(T ), P := Q ∩ R, then we prove that whenever R has a conch maximal subring S with (S : R) = P , then T has a conch maximal subring V such that (V : T ) = Q and V ∩ R = S. It is shown that if K is an algebraically closed field which is not algebraic over its prime subring and R is affine ring over K, then for each prime ideal P of R with ht(P ) ≥ dim(R) − 1, there exists a maximal subring S of R with (S : R) = P . If R is a normal affine integral domain over a field K, then we prove that R is an integrally closed maximal subring of a ring T if and only if dim(R) = 1 and in particular in this case (R : T ) = 0.

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“…The seminal work on FIP and FCP by R. Gilmer is settled for R-subalgebras of K (also called overrings of R), where R is a domain and K its quotient field. In particular, [12,Theorem 2.14] shows that R ⊆ S has FCP for each overring S of R only if R/C is an Artinian ring, where C = (R : R) is the conductor of R in its integral closure. This necessary Artinian condition is not surprisingly present in all our results.…”

Section: Introduction and Notationmentioning

confidence: 99%