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On weakly projective and weakly injective modules

Mohammad Saleh (2004)

Commentationes Mathematicae Universitatis Carolinae

The purpose of this paper is to further the study of weakly injective and weakly projective modules as a generalization of injective and projective modules. For a locally q.f.d. module M , there exists a module K ∈ σ [ M ] such that K ⊕ N is weakly injective in σ [ M ] , for any N ∈ σ [ M ] . Similarly, if M is projective and right perfect in σ [ M ] , then there exists a module K ∈ σ [ M ] such that K ⊕ N is weakly projective in σ [ M ] , for any N ∈ σ [ M ] . Consequently, over a right perfect ring every module is a direct summand of a weakly projective module. For...

Phantom maps and purity in modular representation theory, I

D. Benson, G. Gnacadja (1999)

Fundamenta Mathematicae

Let k be a field and G a finite group. By analogy with the theory of phantom maps in topology, a map f : M → ℕ between kG-modules is said to be phantom if its restriction to every finitely generated submodule of M factors through a projective module. We investigate the relationships between the theory of phantom maps, the algebraic theory of purity, and Rickard's idempotent modules. In general, adding one to the pure global dimension of kG gives an upper bound for the number of phantoms we need...

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