A Note on the Buchsbaum-Rim multiplicity of a parameter module

In this article we prove that the Buchsbaum-Rim multiplicity $e(F\mathrm{/}N)$ of a parameter module $N$ in a free module $F=A^r$ is bounded above by the colength ${\mathrm{\ell }}_A(F\mathrm{/}N)$. Moreover, we prove that once the equality ${\mathrm{\ell }}_A(F\mathrm{/}N)=e(F\mathrm{/}N)$ holds true for some parameter module $N$ in $F$, then the base ring $A$ is Cohen-Macaulay.