A NOTE ON PSEUDO-INJECTIVE MODULES

ABSTRACT

Pseudo-injectivity is a generalization of injectivity. In this paper, we give several properties of pseudo-injective modules, and discuss the question of when a pseudo-injective module is injective or quasi-injective.

Key Words:

• Continuous modules
• CS modules
• Injective
• Pseudo-injective modules
• Quasi-injective modules

(2000) Mathematics Subject Classification:

• 16D50
• 16D80

ACKNOWLEDGMENTS

The author would like to thank Professor S.R. López-Permouth for stimulating discussions and helpful comments. Partially supported by Center of Ring Theory and Applications, Ohio University.

Notes

^aMutually injective modules are also called relatively injective modules (for example, in Dung et al. (1994) and Mohamed and Müller (1990)).

^bIn Dung et al. (1994), quasi-injective modules are called self-injective modules, and quasi-continuous modules are called π-injective modules.

^cFollowing Mohamed and Müller (1990), a family {X _λ}_λ∈Λ of submodules of a module M is called a local direct summand of M if ∑_λ∈Λ X _λ is direct and ∑_λ∈F X _λ is a direct summand of M for every finite subset f ⊆ Λ.

^dThe condition (A ₂) in Mohamed and Müller (1990) is defined as follows: For every choice of x ∈ M _α(α ∈ I) and m _{α i}  ∈ M _{α i} for distant α _i  ∈ I (i ∈ ℕ) such that ann(m _i ) ⊇ ann(x), the ascending sequence ∩ _i≥n ann(m _i ) (n ∈ ℕ) becomes stationary.

^#Communicated by M. Ferrero.