Self-injectivity of $\EuScript{M}(X,\mathcal{A})$ versus $\EuScript{M}(X,\mathcal{A})$ modulo its socle

Abstract

Let $\mathcal{A}$ be a field of subsets of a set $X$ and $\EuScript{M}(X,\mathcal{A})$ be the ring of all real valued $\mathcal{A}$-measurable functions on $X$. It is shown that $\EuScript{M}(X,\mathcal{A})$ is self-injective if and only if $\mathcal{A}$ is a complete and ${\mathfrak{c}}^{+}$- additive field of sets. This answers a question raised in [H. Azadi, M. Henriksen and E. Momtahan, \textit{Some properties of algebras of real valued measurable functions}, Acta Math. Hungar, 124, (2009), 15--23]. Also, it is observed that if $\mathcal{A}$ is a $\sigma$-field, $\EuScript{M}(X,\mathcal{A})$ modulo its socle is self-injective if and only if $\mathcal{A}$ is a complete and ${\mathfrak{c}}^{+}$- additive field of sets with a finite number of atoms.