✦ LIBER ✦

Direct-Sum Decompositions over Local Rings

✍ Scribed by Roger Wiegand

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 125 KB
Volume: 240
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.2000.8657

No coin nor oath required. For personal study only.

✦ Synopsis

Let R, m be a local ring commutative and Noetherian . If R is complete or, . more generally, Henselian , one has the Krull᎐Schmidt uniqueness theorem for direct sums of indecomposable finitely generated R-modules. By passing to the m-adic completion R, we can get a measure of how badly the Krull᎐Schmidt theorem can fail for a more general local ring. We assign to each finitely generated

to the monoid q M of isomorphism classes of modules that are direct summands of direct sums of finitely many copies of M. The main theorem of this paper states that every full submonoid of ‫ގ‬ n arises in this fashion. Moreover, the local ring R realizing a given full submonoid can always be taken to be a two-dimensional unique factorization domain. The theorem has two non-commutative conse-Ž . quences: 1 a new proof of a recent theorem of Facchini and Herbera characterizing the monoid of isomorphism classes of finitely generated projective right Ž . modules over a non-commutative semilocal ring, and 2 a characterization of Ž . the monoids q N , where N is an Artinian right module over an arbitrary ring.

📜 SIMILAR VOLUMES

Direct Sums of Representations as Module

Direct Sums of Representations as Modules over Their Endomorphism Rings

✍ Birge Huisgen-Zimmermann; Manuel Saorı&#x0301;n 📂 Article 📅 2002 🏛 Elsevier Science 🌐 English ⚖ 178 KB

Our investigation into the endo-structure of infinite direct sums i∈ I M i of indecomposable modules M i -over a ring R with identity-is centered on the following question: If S = End R i∈ I M i , how much pressure, in terms of the S-structure of i∈ I M i , is required to force the M i into finitely

SCr Modules over Local Rings

✍ Jee Heub Koh 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 134 KB

We show that the following two conditions, for each integer r G 1, are equivalent for a finitely generated module M over a complete Noetherian local ring Ž . R, ᒊ : embeddable in E r , where E denotes the injective hull of the residue field Rrᒊ. Ž . r Ž . b Either M ; E , or else dim Hom Rrᒊ, M s k

Corings over rings with local units

✍ L. El Kaoutit 📂 Article 📅 2009 🏛 John Wiley and Sons 🌐 English ⚖ 280 KB

## Abstract We show that the bicategory whose 0‐cells are corings over rings with local units is bi‐equivalent to the bicategory of comonads in (right) unital modules whose underlying functors are right exact and preserve direct sums. A base ring extension of a coring by an adjunction is introduced

Dualizing Complexes over Noncommutative

Dualizing Complexes over Noncommutative Local Rings

✍ Q.-S Wu; J.J Zhang 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 225 KB

We prove an existence theorem for dualizing complexes over noncommutative noetherian complete semilocal algebras, which generalizes Van den Bergh's existence theorem in the graded case. Using the dualizing complex, noncommutative versions of Bass theorem and the no-holes theorem are proved. We also

Quadratic Gauss Sums over Finite Commuta

Quadratic Gauss Sums over Finite Commutative Rings

✍ Fernando Szechtman 📂 Article 📅 2002 🏛 Elsevier Science 🌐 English ⚖ 144 KB

Groups Generated by Symplectic Transvect

Groups Generated by Symplectic Transvections over Local Rings

✍ Hiroyuki Ishibashi 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 335 KB

Let R be a commutative local ring with the unique maximal ideal A. Let V be a Ž . free module of rank n over R. And let Sp V be the symplectic group on V with n an alternating bilinear form f : V = V ª R. We study the generation of a subgroup author's conjecture in ''Generators and Relations in Gro