An analogue of the Magnus problem for associative algebras

TL;DR

This paper extends the Magnus theorem to associative algebras without unity, showing that certain algebraic structures with specific generators and relations are free algebras.

Contribution

It provides a new analogue of the Magnus theorem applicable to associative algebras without unity over any field.

Findings

Algebras with n+k generators and k relations are free if they have an n-element generating system.

The theorem applies over arbitrary fields.

It generalizes the classical Magnus theorem to a broader algebraic context.

Abstract

We prove an analogue of the Magnus theorem for associative algebras without unity over arbitrary fields. Namely, if an algebra is given by n+k generators and k relations and has an n-element system of generators, then this algebra is a free algebra of rank n.