Anyone knows a reference for the following result in a commutative ring R?

Let A be an nxn matrix with entries in R. Then, the cardinality of the conjugacy class of A equals the quotient of the cardinality of the set of nxn invertible matrices with entries in R and the cardinality of the centralizer of A.


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    $\begingroup$ The phrase "the quotient" suggests that $R$ is assumed to be finite. The fact you want to prove has little to do with rings and matrices. In any finite group, the cardinality of any conjugacy class $A$ is the quotient of the group order by the order of the centralizer of an element of $A$. This result should be in almost any group theory textbook. $\endgroup$ – Andreas Blass Sep 12 at 0:41