Classification of chain rings

An associative Artinian ring with an identity is a chain ring if its lattice of left (right) ideals forms a unique chain. In this article, we first prove that for every chain ring, there exists a certain finite commutative chain subring which characterizes it. Using this fact, we classify chain rings with invariants $p,n,r,k,k^{\prime },m$ up to isomorphism by finite commutative chain rings ( $k^{\prime }=1$). Thus the classification of chain rings is reduced to that of finite commutative chain rings.