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We will devote this chapter to dualizing the results obtained in Chapter 2. Let M be a module and X be a class of R-modules closed under isomorphisms. A homomorphism p : X → M is an X -precover if any other g : X → M with X ∈ X factorizes through it. And an X -precover is called an X -cover if, moreover, any h : X -→ X such that p • h = p must be an automorphism Definition 5.2 A module M with an X -cover p : X → M is called X -automorphism coinvariant if, for any automorphism g : X → X, there exists an endomorphism f : M → M such that f • p = p • g. Remark 5.3 As in Remark 2.11, the preceding definition can be easily extended to modules having X -precovers. Moreover, if p : X → M is an epimorphic cover, then M is X -automorphism-invariant precisely when the cover p induces a group isomorphism : Aut(M) ∼ = Aut(X)/ coGal(X), where coGal(X) = {g ∈ Aut(X) | p • g = p} is usually called the co-Galois group of the cover p (see e.g. [28, 29]). Definition 5.4 A module M is called a quasi-projective module if, for every submodule N of M, any homomorphism φ : M → M/N can be lifted to a homomorphism ψ : M → M. Note that if X is the class of projective modules over a right perfect ring, then t

Publisher

Cambridge University Press

Published

2021

Language

EN