Consider the subring $k[x^2,x^3]\subseteq k[x]$. Now we may consider the ideal $(x^4)$ in this ring, and quotient so that:

Now this ring is artin local with maximal ideal $\mathfrak{m}=(x^2,x^3)$. We can show that the images in $\mathfrak{m}/{\mathfrak{m}}^2$ are linearly independent. This is an local artin ring which does not satisfy the previous proposition.