On generalization of two results of Foxby

Let $(A,\mathfrak{m})$ be a Noetherian local ring of dimension $d$ and let $M$ be a finitely generated $A$-module. Assume $M$ has rank $r>0$. We show that if $M$ is NOT Cohen-Macaulay then ${\mu }_d(\mathfrak{m},M)>r$. If further $A$ is unmixed and ${\mu }_n(\mathfrak{m},M)\le 1$ for some $n\ge d$ then we prove $\text{injdim }M<\mathrm{\infty }$ and $A$ is Cohen-Macaulay.