faithful modules over a finite dimensional commutative algebra

Question

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 $dim_k(M)\geq dim_k(A)$?

Answer 1

$A$ 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 $3$.