Diagonalizable matrix

Let \(R\) be a commutative ring and \(A\) an \(n\times n\) matrix over \(R\). \(A\) is diagonalizable if it is similar to a diagonal matrix: there exist a diagonal \(n\times n\) matrix \(D\) over \(R\) and \(P\in \GL_n(R)\) such that \(A = PDP^{-1}\).