On classification of finite commutative chain rings

Let $R$ be a finite commutative chain ring with invariants $p,n,r,k,m.$ It is known that $R$ is an extension over a Galois ring $GR(p^n,r)$ by an Eisenstein polynomial of some degree $k$. If $p\nmid k,$ the enumeration of such rings is known. However, when $p\mid k$, relatively little is known about the classification of these rings. The main purpose of this article is to investigate the classification of all finite commutative chain rings with given invariants $p,n,r,k,m$ up to isomorphism when $p\mid k.$ Based on the notion of j-diagram initiated by Ayoub, the number of isomorphism classes of finite (complete) chain rings with $(p-1)\nmid k$ is determined. In addition, we study the case $(p-1)\mid k,$ and show that the classification is strongly dependent on Eisenstein polynomials not only on $p,n,r,k,m.$ In this case, we classify finite (incomplete) chain rings under some conditions concerning the Eisenstein polynomials. These results yield immediate corollaries for p-adic fields, coding theory and geometry.