I'm looking for a reference of the following statement (which can easily be proved by Laplace's formula and induction):

Let $R$ be a commutative ring with identity and let $A$ be an invertible matrix over $R$. Then there is a permutation matrix $P$ such that the diagonal of $PA$ has no zero.

Edit: Of course, $R$ has to be a domain to make the arguments (formula of Leibniz or Laplace) work. Futhermore, it's sufficient to require $det(A) \neq 0$ (instead of $A$ being invertible).