I understand what the invariant factors of a finitely generated module over an ED are, but what does it mean to ask, say, for the invariant factors of a matrix over $\mathbb{R}$?
I cannot find a clear source for this online. Apparently they are the invariant factors in the smith normal form of $M-xI$ in $R[X]$ but I don’t see how they relate to invariant factors of a module.