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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

##### MSC:
 13B99 Commutative ring extensions and related topics 13C13 Other special types of modules and ideals in commutative rings
##### Keywords:
progenerator modules; morphisms; homotopy; injective; completion
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##### References:
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