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Local morphisms and modules with a semilocal endomorphism ring

Alberto Facchini·Dolors Herbera
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Abstract

An associative ring with 1 is said to be semilocal provided it is semisimple artinian modulo its Jacobson radical, that is, modulo its Jacobson radical it is isomorphic to a finite product of matrices over division rings. Modules with a semilocal endomorphism ring inherit some properties of semisimple modules. For example, they can be decomposed into a finite direct sum of indecomposable submodules, such decomposition is not unique but there are only a finite number of different ones, they cancel from direct sums and they satisfy the n-th root uniqueness property. A ring homomorphism is said to be local if it carries non-units to non-units. Semilocal rings can be characterized as those rings having a local homomorphism to a semisimple artinian ring. In this paper show that local homomorphisms appear frequently in module theory. As a consequence it will follow that many interesting classes of modules, as for example finitely presented modules over semilocal rings, have a semilocal endomorphism ring.

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Alberto Facchini, Dolors Herbera. 2005-10-05. Local morphisms and modules with a semilocal endomorphism ring. https://arxiv.org/abs/math/0510104

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