Decompositions of injective modules

Warfield, R. B.

doi:10.2140/pjm.1969.31.263

Cited by 58 publications

(23 citation statements)

References 7 publications

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“…Since the property of being an exchange ring has a description in terms of the solvability of equations (Proposition 20(d)), it follows that E is an exchange ring, so A has the finite exchange property. The remaining statements follow from isomorphic refinement: [1, Theorem 7.1] or [11,Theorem 7]. [] In the case where R = S and S is von Neumann regular, this solves a problem posed by Pierce [6, Question 5, p. 109].…”

Section: The Main Theoremsmentioning

confidence: 99%

Algebras over zero-dimensional rings

Goodearl

WarfieldJr.

1976

Math. Ann.

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The main results of this paper are cancellation, n-th root, and isomorphic refinement theorems for finitely generated modules over algebras over commutative zero-dimensional rings. The cancellation and n-th root properties are established more generally for algebras over commutative rings which are von Neumann regular modulo their radicals. Analogues of the theorems of Serre and ForsterSwan are also obtained.Suppose that R is a ring which is either (a) an algebra over a zero-dimensional commutative ring S, or (b) a locally module-finite algebra over a commutative ring S which is von Neumann regular modulo its radical. In either case, it is shown that (i) if A and B are finitely presented R-modules and if for every maximal ideal M of S, AM --BM (as R M modules) then A ~ B, and (ii) irA and B are finitely generated with A finitely presented, then there is an epimorphism A~B if there is one locally. This implies analogues of the theorems of Serre and Forster-Swan in these situations. In case (b), it is also shown that (iii) (the cancellation property) if A is a finitely generated left R-module and B and C are arbitrary left R-modules such that AOB~AOC, then B~C, and (iv) (n-th root property) if A is a finitely generated R-module and A" ~ B", then A ~ B. If both (a) and (b) hold (so that R is a locally module-finite algebra over a zero-dimensional ring), then we obtain (v) an isomorphic refinement theorem, which is as close as one can hope to come to the Krull-Schmidt theorem in the non-Noetherian situation: any two direct sum decompositions of a finitely generated R-module have isomorphic refinements. (i)-(iv) hold, in particular, for modules over commutative quasi-local rings, and finite algebras over these rings, and even in this situation they were previously known only when the rings were Noetherian. (v) does not hold in that generality, but even in the case when R = S and S is commutative and von Neumann regular the resialt is new.All rings and algebras in this paper are associative with t, and all modules are unital.

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Section: The Main Theoremsmentioning

confidence: 99%

Algebras over zero-dimensional rings

Goodearl

WarfieldJr.

1976

Math. Ann.

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“…Hence N = lj © M' a © (N n Af*) and N n M* is a maximal submodule of A/*. M* is also a direct sum of submodules isomorphic to A/, by [16]. Therefore \I -J\-1 by the above and the maximality of lj © M' a .…”

Section: Lifting Property Of Injectivesmentioning

confidence: 84%

On maxi-quasiprojective modules

Harada

1983

J Aust Math Soc A

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We have defined a mini-injective module and given some structures of self mini-injective rings and certain relationships between such rings and QF-rings in [8] and [9].In this short note we shall study the modules dual to mini-injective modules, which we call maxi-quasiprojective modules. We shall give a characterization and some structures, in terms of the above modules, of those rings whose every injective module has the lifting property of direct decompositions modulo the Jacobian radical (see [5], [6] and [7]). Furthermore, we shall show that the above rings are closely related to QF-rings (see [8] Throughout this note, we assume that a ring R contains an identity and every module is a unitary right /^-module. We always assume that R is a right artinian ring unless otherwise stated. However, some of the first part of this note is valid without this assumption.A part of this paper was prepared when the author visited University of Sydney in 1981. The author would like to express his thanks to Professor M. Kelly and his colleagues for their kind hospitality, and to the referee for the useful suggestion to revise the paper.

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“…Here ⊕ λ∈ M λ denotes the direct sum of the modules in the family and π λ denotes the canonical projection of the direct sum onto M λ (see [48] or [83]). We say that the R-module U is quasismall if whenever U is isomorphic to a direct summand of a direct sum ⊕ λ∈ M λ of right R-modules M λ , there is a finite subset F of such that U is isomorphic to a direct summand of ⊕ λ∈F M λ .…”

Section: Quasismall Modules and Weak Krull-schmidt For Infinite Direcmentioning

confidence: 99%

Direct-sum decompositions of modules with semilocal endomorphism rings

Facchini

2012

Bull. Math. Sci.

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According to the classical Krull-Schmidt Theorem, any module of finite composition length decomposes as a direct sum of indecomposable modules in an essentially unique way, that is, unique up to isomorphism of the indecomposable summands and a permutation of the summands. Modules that do not have finite composition length can have completely different behaviors. In this survey, we consider in particular the case of the modules M R whose endomorphism ring E := End(M R ) is a semilocal ring, that is, E/J (E) is a semisimple artinian ring. For instance, modules of finite composition length have a semilocal endomorphism ring, but several other classes of modules also have a semilocal endomorphism ring, for example artinian modules, finite direct sums of uniserial modules, finitely generated modules over commutative semilocal rings, and finitely presented modules over arbitrary semilocal rings. Several interesting phenomena appear in these cases. For instance, modules with a semilocal endomorphism ring have very regular direct-sum decompositions into indecomposables, their direct summands can be described via lattices, and direct-sum decompositions into indecomposables (=uniserial submodules) of finite direct sums of Communicated by S.K. Jain.Partially supported by Università di Padova (Progetto di ricerca di Ateneo CPDA105885/10 "Differential graded categories" and Progetto ex 60% "Anelli e categorie di moduli").

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Decompositions of injective modules

Cited by 58 publications

References 7 publications

Algebras over zero-dimensional rings

Algebras over zero-dimensional rings

On maxi-quasiprojective modules

Direct-sum decompositions of modules with semilocal endomorphism rings

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