  In this paper we prove that if $R$ is a commutative refinement ring and $M$,
$N$ are two $R$-modules then, $M\cong N$ if and only if for every maximal ideal
$m$ of $R$, $M_m\cong N_m$. We prove if $R$ is a refinement ring, then every
regular matrix over $\frac{R}{J(R)}$ admits a diagonal reduction iff every
regular matrix over $R$ admits a diagonal reduction.
