Let $R$ be a finite commutative ring with identity and $R=R_1+...+R_k$ its decomposition in a direct sum of finite local rings. Considere the matrix ring $M_n(R)$ of $n\times n$ matrices with elements in $R$. It is easy to see that a matrix $A$ decomposes in a unique way as
$$A=A_1+ .... +A_k,$$
with $A_i\in R_i$. Let $per(A)$ denote the permanent of the matrix $A$.
Is true that $per(A)$ is a unit in $R$ if and only if each $per(A_i)$ is a unit in $A_i$?