Journal Article10.1017/S1446788717000313
On modules over commutative rings
László Fuchs,Sang Bum Lee +1 more
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TL;DR: In this paper, the authors extend several results of interest that have been proved for modules over integral domains to modules over arbitrary commutative rings with identity, where the classical ring of quotients of will play the role of the field of quotient when zero-divisors are present.
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Abstract: Our main purpose is to extend several results of interest that have been proved for modules over integral domains to modules over arbitrary commutative rings with identity. The classical ring of quotients of will play the role of the field of quotients when zero-divisors are present. After discussing torsion-freeness and divisibility (Sections 2–3), we study Matlis-cotorsion modules and their roles in two category equivalences (Sections 4–5). These equivalences are established via the same functors as in the domain case, but instead of injective direct sums one has to take the full subcategory of -modules into consideration. Finally, we prove results on Matlis rings, i.e. on rings for which has projective dimension (Theorem 6.4).
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Citations
Almost perfect commutative rings
László Fuchs,Luigi Salce +1 more
TL;DR: In this paper, almost perfect commutative rings with zero divisors of zero are defined as an extension of a T-nilpotent ideal by a subdirect product of a finite number of almost perfect domains.
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C*-algebras of Clifford semigroups
J. Duncan,Alan L. T. Paterson +1 more
TL;DR: In this paper, the authors investigated the representation theory of a Clifford semigroup and showed that it is determined by an enveloping Clifford semiigroup UC(S) = ∪ {Gx: x ∈ X} where X is the filter completion of the semilattice E. The representation theory was described in terms of disintegration theory and sheaf theory.
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When weak-injective modules decompose like injectives
TL;DR: In this article, it was shown that a very few commutative rings R possess the property that their weak-injective modules admit (up to isomorphism) unique decompositions into direct sums of indecomposable modules each of which is the injective envelope of a cyclic module with a prime ideal.
1
Weak-injective modules over commutative rings
TL;DR: Weak-injective modules were defined as modules M for which ExtR1(N,M)=0 holds for all modules N of weak dimension ≤ 1 in this paper, and they were studied over integral domains in several papers.
1
A note on Matlis localizations
Xiaolei Zhang
TL;DR: S-Matlis rings are characterized in terms of S-strongly flat, S-weakly cotorsion and S-h-divisible modules.
References
Finitistic dimension and a homological generalization of semi-primary rings
TL;DR: In this article, Kaplansky showed that a commutative ring R is left T-nilpotent if, given any sequence {at} of elements in N, there exists an re such that ai • • • an = 0.
•Book
Approximations and Endomorphism Algebras of Modules
Rüdiger Göbel,Jan Trlifaj +1 more
- 01 Jan 2006
TL;DR: Approximations and endomorphism algebras of modules have been studied extensively in the literature since 2006 as mentioned in this paper, with a focus on the impossibility of classification for modules over general rings.
627
All modules have flat covers
TL;DR: Two different proofs that the flat cover conjecture is true: that is, every module has a flat cover are given, and each has a model‐theoretic flavour.
453
Divisible modules and localization
TL;DR: Fuchs and Salce as discussed by the authors showed that if R is a Matlis domain any submodule of Q/R can be embedded in a countably generated direct sum of quotient fields.
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