Mixed Injective Modules

Abstract

Abstract. Since Azumaya introduced the notion of A-injectivity in 1974, several generalizations have been investigated by a number of authors. We introduce some more generalizations and discuss their connection to the previous ones.2000 Mathematics Subject Classification. Primary 16D50, Secondary 16D80. B is almost A-injective if for any ϕ : A ≥ X −→ B, there exists a homomorphism ϕ 1 : A −→ B that extends ϕ (injectivity behaviour), or there exists a non-zero summand A 2 ≤ A and a homomorphism ϕ 2 : B −→ A 2 such that ϕ 2 ϕ = π A 2 | X , where π A 2 is the projection of A onto A 2 (which we refer to as opposite injectivity behaviour). We note that for an indecomposable module A, we have that B is almost A-injective if and only if for any ϕ : A ≥ X −→ B, there exists a homomorphism ϕ 1 : A −→ B or a homomorphism ϕ 2 : B −→ A such that the following diagrams commute: