I know that if I have a matrix $A,$ the characteristic polynomial is determinant of the matrix $(A-\lambda I)$ where, $\lambda$ is an eigenvalue and $I$ is an identity matrix, and the characteristic equation is the characteristic polynomial equated to zero.
Let $k$ be a field and $R = k[x]$, I want to find the characteristic polynomial of an $R$-module of order (product of its prime ideals or product of all elementary divisors/ invariant factors) is $(x-1)^3(x + 1)^2.$ but I do not know where is the matrix in my case? where is the eigenvalue?
Could anyone help me answer those questions, please?