Left ideals in matrix rings over finite fields

TL;DR

This paper investigates the structure of left ideals in matrix rings over finite fields, providing formulas and explicit generators for counting and constructing these ideals based on their rank.

Contribution

It introduces explicit methods to count and generate all left ideals in matrix rings over finite fields, including the enumeration of idempotent generators.

Findings

Number of left ideals computed

Count of idempotent generators determined

Explicit generators for ideals of each rank provided

Abstract

It is well-known that each left ideals in a matrix rings over a finite field is generated by an idempotent matrix. In this work we compute the number of left ideals in these rings, the number of different idempotents generating each left ideal, and give explicitly a set of idempotent generators of all left ideals of a given rank.