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.