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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 $A_i$?

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 The elements of $R$ are tuples $(r_1,...,r_n)$, $r_i \in R_i$ with componentwise addition and multiplication. Since the permanent is defined by addition and multiplication, $per(A) = (per(A_1),...,per(A_n))$ holds. Eventually $R^\times = \prod_i R_i^\times$ implies $per(A) \in R^\times$ iff $per(A_i) \in R_i^\times$ for $i=1,...,n$. – Ralph Mar 10 2012 at 15:57

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