Enumeration of Finite Rings with Jacobson Radical of Cube Zero

TL;DR

This paper classifies finite associative rings with identity where the Jacobson radical is of cube zero, providing invariants and counting isomorphism classes for rings with property(T).

Contribution

It introduces invariants for rings with property(T) and determines the number of isomorphism classes in certain cases, extending prior classifications.

Findings

Invariants for rings with property(T) are established.

Number of isomorphism classes is determined in specific cases.

Rings with Jacobson radical of cube zero are fully characterized in these contexts.

Abstract

In [1], finite associative rings wih identity and such that the set of all zero-divisors form and ideal M, called the Jacobson Radical, of cube zero and square non-zero, were constructed for all the characteristics. These rings are completely primary and we call them rings with property(T). In this paper, we associate with each ring with property(T) and characteristic p, invariants (integers) and determine (in certain cases) the number of isomorphism classes of these rings with given invariants.