Finite Commutative Chain Rings

A commutative ring with identity is called a chain ring if all its ideals form a chain under inclusion. A finite chain ring, roughly speaking, is an extension over a Galois ring of characteristic pⁿusing an Eisenstein polynomial of degree k. When p∤k, such rings were classified up to isomorphism by Clark and Liang. However, relatively little is known about finite chain rings when p∣k. In this paper, we allowed p∣k. When n=2 or when p∥k but (p−1)∤k, we classified all pure finite chain rings up to isomorphism. Under the assumption that (p−1)∤k, we also determined the structures of groups of units of all finite chain rings.