Quasi--bases for Modules over a Commutative Ring

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 $(\Omega,\mathcal{F},P)$.