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

## TL;DR

This paper characterizes when the ring of real-valued measurable functions is self-injective, linking this property to the completeness and additivity of the underlying field of sets, and explores the structure modulo its socle.

## Contribution

It provides a complete characterization of self-injectivity for rings of measurable functions based on properties of the underlying set algebra, answering a previously posed question.

## Key findings

- Self-injectivity of $ ext{M}(X, ext{A})$ is equivalent to $ ext{A}$ being complete and $rak{c}^+$-additive.
- Modulo its socle, $ ext{M}(X, ext{A})$ is self-injective if and only if $ ext{A}$ is complete, $rak{c}^+$-additive, and has finitely many atoms.
- The results resolve an open question about the algebraic properties of rings of measurable functions.

## 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.