Homomorphisms of progenerator modules under a change of base ring. (English) Zbl 0638.13008

This paper is concerned with the behaviour of morphisms under a certain functor \({\mathcal M}\) from the category of commutative rings to the category of abelian monoids introduced by F. Demeyer and the author [“Homomorphisms of progenerator modules”, J. Algebra 113, No.2, 379-398 (1988)]. For a given commutative ring, \({\mathcal M}(R)\) consists of the equivalence classes of an equivalence relation, called homotopy, defined on the set of all R-homomorphisms between progenerator R-modules M and N.
A typical question asked is, given a ring homomorphism \(\sigma: R\to S,\) when is the monoid homomorphism \({\mathcal M}(\sigma)\) injective? One theorem lists necessary conditions for this to occur, for example that \(\sigma\) is injective and that, for ideals I and J of R, \(SI=SJ\) implies \(I=J\), while another one lists sufficient conditions, for example if S is the completion of a local ring R at the maximal ideal and \(\sigma\) is the natural map.
Reviewer: D.A.Jordan


13B99 Commutative ring extensions and related topics
13C13 Other special types of modules and ideals in commutative rings
Full Text: DOI


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