共查询到20条相似文献,搜索用时 328 毫秒
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Saeed Nasseh Sean Sather-Wagstaff Ryo Takahashi Keller VandeBogert 《Journal of Pure and Applied Algebra》2019,223(3):1272-1287
We construct a local Cohen–Macaulay ring R with a prime ideal such that R satisfies the uniform Auslander condition (UAC), but the localization does not satisfy Auslander's condition (AC). Given any positive integer n, we also construct a local Cohen–Macaulay ring R with a prime ideal such that R has exactly two non-isomorphic semidualizing modules, but the localization has non-isomorphic semidualizing modules. Each of these examples is constructed as a fiber product of two local rings over their common residue field. Additionally, we characterize the non-trivial Cohen–Macaulay fiber products of finite Cohen–Macaulay type. 相似文献
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Christian U. Jensen Søren Jøndrup Anders Thorup 《Journal of Pure and Applied Algebra》2019,223(5):1869-1896
Classical results concerning slenderness for commutative integral domains are generalized to commutative rings with zero divisors. This is done by extending the methods from the domain case and bringing them in connection with results on the linear topologies associated to non-discrete Hausdorff filtrations. In many cases a weakened notion “almost slenderness” of slenderness is appropriate for rings with zero divisors. Special results for countable rings are extended to rings said to be of “bounded type” (including countable rings, ‘small’ rings, and, for instance, rings that are countably generated as algebras over an Artinian ring).More precisely, for a ring R of bounded type it is proved that R is slender if R is reduced and has no simple ideals, or if R is Noetherian and has no simple ideals; moreover, R is almost slender if R is not perfect (in the sense of H. Bass). We use our methods to study various special classes of rings, for instance von Neumann regular rings and valuation rings. Among other results we show that the following two rings are slender: the ring of Puiseux series over a field and the von Neumann regular ring over a von Neumann regular ring k.For a Noetherian ring R we prove that R is a finite product of local complete rings iff R satisfies one of several (equivalent) conditions of algebraic compactness. A 1-dimensional Noetherian ring is outside this ‘compact’ class precisely when it is almost slender. For the rings of classical algebraic geometry we prove that a localization of an algebra finitely generated over a field is either Artinian or almost slender. Finally, we show that a Noetherian ring R is a finite product of local complete rings with finite residue fields exactly when there exists a map of R-algebras vanishing on . 相似文献
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Mi Hee Park 《Journal of Pure and Applied Algebra》2019,223(9):3980-3988
There are many Noetherian-like rings. Among them, we are interested in SFT-rings, piecewise Noetherian rings, and rings with Noetherian prime spectrum. Some of them are stable under polynomial extensions but none of them are stable under power series extensions. We give partial answers to some open questions related with stabilities of such rings. In particular, we show that any mixed extensions over a zero-dimensional SFT ring R are also SFT-rings, and that if R is an SFT-domain such that is integrally closed for each prime ideal P of R, then is an SFT-ring. We also give a direct proof that if R is an SFT Prüfer domain, then is an SFT-ring. Finally, we show that the power series extension over a Prüfer domain R is piecewise Noetherian if and only if R is Noetherian. 相似文献
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Raymond C. Heitmann 《Journal of Pure and Applied Algebra》2022,226(1):106809
It is shown that if is a complete local domain with and is its integral closure in an algebraic closure of the quotient field, then both the -adic and p-adic completions of are integral domains. More generally, this theorem remains true if the completeness assumption is relaxed to allow R to be an analytically irreducible Henselian local ring. It is also shown that these rings, which are Cohen-Macaulay R-modules (even balanced in the -adic case), will have dimension larger than the dimension of R unless . 相似文献
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Bhargav Bhatt 《Journal of Pure and Applied Algebra》2019,223(5):1940-1945
Given a commutative ring A and a finitely generated ideal I, we prove that -torsion A-modules that are also I-adically complete (or merely derived I-complete) must have bounded -torsion, i.e., they are killed by for some . 相似文献
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For an ideal generated by all square-free monomials of degree m in a polynomial ring R with n variables, we obtain a specific embedding of a canonical module of to itself. The construction of this explicit embedding depends on a minimal free R-resolution of an ideal generated by . Using this embedding, we give a resolution of connected sums of several copies of certain Artin -algebras where is a field. 相似文献
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Md. Ali Zinna 《Journal of Pure and Applied Algebra》2019,223(2):783-793
Let R be a commutative Noetherian ring of dimension two with and let . Let P be a projective A-module of rank 2. In this article, we prove that P is cancellative if is cancellative. 相似文献
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《Journal of Pure and Applied Algebra》2022,226(4):106890
A commutative Noetherian ring R is said to be Tor-persistent if, for any finitely generated R-module M, the vanishing of for implies M has finite projective dimension. An open question of Avramov, et al. asks whether any such R is Tor-persistent. In this work, we exploit properties of exterior powers of modules and complexes to provide several partial answers to this question; in particular, we show that every local ring with is Tor-persistent. As a consequence of our methods, we provide a new proof of the Tachikawa Conjecture for positively graded rings over a field of characteristic different from 2. 相似文献
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Let C be a chain complex of finitely generated free modules over a commutative Laurent polynomial ring in s indeterminates. Given a group homomorphism we let denote the resulting induced complex over the Laurent polynomial ring in t indeterminates. We prove that the Betti number jump loci, that is, the sets of those homomorphisms p such that , have a surprisingly simple structure. We allow non-unital commutative rings of coefficients, and work with a notion of Betti numbers that generalises both the usual one for integral domains, and the analogous concept involving McCoy ranks in case of unital commutative rings. 相似文献
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The aim of this paper is to give a characterization of path connected topological fields, inspired by the classical Gelfand correspondence between a compact Hausdorff topological space X and the space of maximal ideals of the ring of real valued continuous functions . More explicitly, our motivation is the following question: What is the essential property of the topological field that makes such a correspondence valid for all compact Hausdorff spaces? It turns out that such a perfect correspondence exists if and only if F is a path connected topological field. 相似文献
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Let R be a polynomial ring over a field and I an ideal generated by three forms of degree three. Motivated by Stillman's question, Engheta proved that the projective dimension of is at most 36, although the example with largest projective dimension he constructed has . Based on computational evidence, it had been conjectured that . In the present paper we prove this conjectured sharp bound. 相似文献
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Let R be an affine domain of dimension over a field of characteristic 0 and . Let be a local complete intersection ideal of height n such that . This paper examines under what condition I is surjective image of a projective D-module of rank n. 相似文献
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Let R and S be standard graded algebras over a field k, and and homogeneous ideals. Denote by P the sum of the extensions of I and J to . We investigate several important homological invariants of powers of P based on the information about I and J, with focus on finding the exact formulas for these invariants. Our investigation exploits certain Tor vanishing property of natural inclusion maps between consecutive powers of I and J. As a consequence, we provide fairly complete information about the depth and regularity of powers of P given that R and S are polynomial rings and either or I and J are generated by monomials. 相似文献