Let $A$ be a commutative Frobenius algebra over a field $K$ (we can assume that $A$ is local). We can assume $A=K[x_1,...,x_r]/I$ for an ideal $I$ with $J^n \subseteq I \subseteq J^2$ where $J=<x_i>$, the ideal of $K[x_1,...,x_r]$.

Let $B=\{v_i \}$ be a monomial vector space basis (meaning $v_i$ is a homogeneous polynomial in $x_i$) of $A$ containing the unit of $A$. Let $M_i:=v_i A$ and $M:= \bigoplus_{}^{}{M_i}$ and $C:=\underline{End_A}(M)$ the stable endomorphism ring of $M$.

Question: Is $C$ independent of the choosen basis $B$ up to isomorphism?