Abstract
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 inverted iota k, the enumeration of such rings is known. However, when p vertical bar 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 vertical bar k. Based on the notion of j-diagram initiated by Ayoub, the number of isomorphism classes of finite (complete) chain rings with (p - 1) inverted iota k is determined. In addition, we study the case (p - 1) vertical bar 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.