Quasi--bases for Modules over a Commutative Ring

TL;DR

This paper introduces the concept of quasi-bases for modules over commutative rings, establishing conditions for their existence and demonstrating that finitely generated modules over a specific algebra of random variables possess quasi-bases.

Contribution

It defines quasi-bases for modules over commutative rings and proves their existence in finitely generated modules over the algebra of equivalence classes of random variables.

Findings

Quasi-bases are defined for modules over commutative rings.

Existence of quasi-bases is guaranteed under certain properties.

Finitely generated modules over $L^{0}(cal,F,K)$ have quasi-bases.

Abstract

In this paper we present the definition of quasi-bases for modules over a ring that is commutative but not necessarily division and discuss properties that guarantee the existence of quasi-bases. Based on this result we further prove that every finitely generated module over $L^{0}(\mathcal{F},K)$ has a quasi-basis, where $K$ is the scalar field of real numbers or complex numbers and $L^{0}(\mathcal{F},K)$ is the algebra of equivalence classes of $K$--valued random variables defined on a probability space $(\Omega,\mathcal{F},P)$.