Modules with perfect decompositions

Abstract

It is well known that a module M over an arbitrary ring admits an indecomposable decomposition whenever it has the property that every local direct summand of M is a direct summand [28]. Recently, J. L. Gómez Pardo and P. Guil Asensio [18] have shown that requiring this property not only for M but for any direct sum M (ℵ) of copies of M even yields the existence of a decomposition of M in modules with local endomorphism ring which, moreover, satisfies many nice properties of decompositions studied in the literature, like the exchange property, or the property of complementing direct summands. More precisely, it turns out that all these properties coincide if, instead of considering a single module M, we pass to the category Add M of all direct summands of direct sums of copies of M.In the present paper, we continue the investigation of these modules calling them modules with perfect decompositions. In Section 1, we show that a module M has a perfect decomposition if and only if for every direct system (M i , f ji ) I of modules in Add M indexed by a totally ordered set I , the canonical epimorphism π : i∈I M i −→ lim − → M i is a split epimorphism. This allows to shed a new light on a number of known examples of modules with perfect decomposition.The remaining sections are devoted to the role played in this context by certain finiteness conditions over the endomorphism ring S = End M. In fact, every module with a perfect decomposition is S-coperfect, that is, it satisfies the descending chain condition on cyclic S-submodules. Actually, in Section 2, we even show that M is -coperfect over S, i.e. any direct sum M (ℵ) of copies of M is S-coperfect.We thus discuss whether the converse implication also holds true. The best answer that we can give in full generality is the following (see Section 3): -coperfectness over the endomorphism ring implies that the pure epimorphism * The second author thanks the D.G.I. of the Spanish Ministry of Science and Technology and the Fundación "Séneca" of Murcia for their financial support.

Decomposition properties of strict Mittag-Leffler modules

“…We recall that a local direct summand of a module L is a submodule which is the sum of an independent family {N i : i ∈ I } of direct summands of L such that, for any J ⊆ I finite, the sum j ∈J N j is a direct summand. This property, and others equivalent to it, appears in [2,3,11] and its references. The results of these papers concerning this property are summarized in the following one (see [ …”

“…Then X is flat by Proposition 1.5 and, by [15, 51.10], there exists a Q ∈ R-Mod with X ∼ = Hom(M, Q). Recall that a module M is said to have a perfect decomposition (see [3]) if every object in Add M has a decomposition that complements direct summands (see [1, §12]). It is proved in [1,Theorem 29.5] that a finitely generated module has a perfect decomposition if and only if its endomorphism ring is left perfect.…”

One Dimensional Tilting Modules are of Finite Type

“…We reformulate the descending chain condition of Theorem 3.7 (2). We follow the spirit of [22, Theorems 1.9 and 2.1] and of [18,Lemma 3.1], where this type of statement is used to give a description of countably generated projective modules.…”

“…It is inspired by the characterization of the case when C n = M n and γ the identity, that is when φ has a left inverse. The argument in the proof, that we repeat for completeness' sake, was used first by Bass [6] for the case C n = M n = R, later it was completed and extended to more general situations in work by Zimmermann [24], Whitehead [22], Azumaya [5], Angeleri-Hügel and Saorin [2]. …”

Locally Finitely Generated Modules Over Rings With Enough Idempotents

“…We generalize some (classical) results known for rings with identity. Following [4], we say that a decomposition X = i∈I X i such that End R (X i ) is local for each i ∈ I, is a perfect decomposition if any direct sum of copies of i∈I X i complements direct summands (in fact, there are several equivalent conditions due to work of many authors [3,4,12,15,16,19,21], see Subsec. 2.2).…”