# faithful modules over a finite dimensional commutative algebra

Let $$A$$ be a commutative algebra over a field $$k$$ which is finite dimensional as a vector space over $$k$$. Let $$M$$ be a faithful $$A$$-module. Does it follow that $$dim_k(M)\geq dim_k(A)$$?

$$A$$ is a product of local Artinian rings, so the question is local. The best result I know is in this paper of Gulliksen, who proved that if the socle dimension is at most $$3$$, then the length of any faithful module is at least the length of the ring. So the answer is yes if $$A$$ is the product of local Artin rings of socle dimension at most $$3$$. He also gave counter example when the socle dimension is bigger than $$3$$.