The Structure of Local Rings with Singleton Basis and Their Enumeration

A local ring is an associative ring with unique maximal ideal. We associate with each Artinian local ring with singleton basis four invariants (positive integers) p,n,s,t. The purpose of this article is to describe the structure of such rings and classify them (up to isomorphism) with the same invariants. Every local ring with singleton basis can be constructed over its coefficient subring by a certain polynomial called the associated polynomial. These polynomials play significant role in the enumeration.