Decompositions of algebraically compact modules

Facchini, Alberto

doi:10.2140/pjm.1985.116.25

Cited by 4 publications

(2 citation statements)

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“…9]) that if M is a pure-injective right module (over any ring) with the endomorphism ring S, then S = S/Jac(S) (the factor of S modulo its Jacobson radical) is a von Neumann regular right self-injective ring, and idempotents are lifted modulo Jac(S). What follows (see [6]) is that the direct sum decompositions of M are essentially the same as the decompositions of S as a right module over itself. But the direct sum decompositions of right selfinjective von Neumann regular rings are well understood (see [8]).…”

Section: Discussionmentioning

confidence: 98%

How to construct a ‘concrete’ superdecomposable pure-injective module over a string algebra

Puninski¹

2008

Journal of Pure and Applied Algebra

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We construct an element in a direct product of finite dimensional modules over a string algebra such that the pure-injective envelope of this element is a superdecomposable module.A nonzero module M is said to be superdecomposable if M has no indecomposable direct summands. For example, let R be the endomorphism ring of a countable dimensional vector space V , and let I be the ideal of R consisting of all endomorphisms with finite dimensional images. If R = R/I and e is a projection on a subspace of V whose image and coimage are infinite dimensional, then R ∼ = e R ⊕ (1 − e)R as a right module over itself and R ∼ = e R ∼ = (1 − e )R . Furthermore, every nontrivial decomposition of R as a right module is of this form; therefore R is superdecomposable.For more examples, let R = k X, Y be a free algebra over a field k, and let E = E(R R ) be the injective envelope of R considered as a right module over itself. It is easily verified (see [11, Prop. 8.36]) that R R has no nonzero uniform submodules. Therefore the same is true for E, and hence E is a superdecomposable injective module.More generally, this is a common feature of finite dimensional wild algebras, that they usually (conjecturally always) have a superdecomposable pure-injective module (see [11, Ch. 8] for a list of existing results). Here a module M over a finite dimensional algebra A is pure-injective if M is a direct summand of a direct product of finite dimensional A-modules.If A is a tame finite dimensional algebra over a field, it has been believed for a while (see [24, p. 38]) that every pure-injective A-module has an indecomposable direct summand. But recently, Puninski [19] showed that every nondomestic string algebra over a countable field has a superdecomposable pure-injective module (note that every string algebra is tame). The main drawback of his proof is that it depends on the cardinality of the ground field and is highly nonconstructive. Specifically, it is based on a quite ingenious construction of Ziegler [26] which seems to work only for countable fields.

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