On the number of generators of an algebra

TL;DR

This paper generalizes Forster's theorem on module generation to arbitrary finite algebras, including non-unital, non-commutative, and non-associative cases, establishing bounds on the number of generators needed.

Contribution

It extends classical module generation results to a broad class of finite algebras, providing new bounds and a unified framework.

Findings

Generalization of Forster's theorem to arbitrary finite algebras

Applicable to non-unital, non-commutative, and non-associative algebras

Establishes bounds on the number of generators needed

Abstract

A classical theorem of Forster asserts that a finite module $M$ of rank $\leq n$ over a Noetherian ring of Krull dimension $d$ can be generated by $n + d$ elements. We prove a generalization of this result, with "module" replaced by "algebra". Here we allow arbitrary finite algebras, not necessarily unital, commutative or associative. Forster's theorem can be recovered as a special case by viewing a module as an algebra where the product of any two elements is $0$.