Similarity invariants for matrices over a commutative Artinian chain ring

Suppose that $R$ is a commutative Artinian chain ring, $A$ is an $m\times m$ matrix over $R$, and $S$ is a discrete valuation ring such that $R$ is a homomorphic image of $S$. We consider $m$ ideals in the polynomial ring over $S$ that are similarity invariants for matrices over $R$, i.e., these ideals coincide for similar matrices. It is shown that the new invariants are stronger than the Fitting invariants, and that new invariants solve the similarity problem for $2\times 2$ matrices over $R$.