Self-injectivity of versus modulo its socle

Abstract

Let be a field of subsets of a set X and be the ring of all real valued -measurable functions on X. If is closed under arbitrary unions, the self-injectivity of are given by Azadi-Henriksen-Momtahan in 2009. In this article, we improve the latter result and show that is self-injective if and only if is a complete and - additive field of sets. Finally, it is observed that if is a σ-field, modulo its socle is self-injective if and only if is a complete and - additive field of sets with a finite number of atoms.