Some embedding results for associative algebras

TL;DR

This paper explores conditions and methods for embedding associative algebras into larger algebras generated by specific elements or subalgebras, advancing understanding of algebraic embeddings.

Contribution

It introduces new embedding constructions using two subalgebras, combining previous approaches based on generators and subalgebras.

Findings

Established new embedding conditions for associative algebras.

Merged existing methods to create embeddings using two subalgebras.

Posed questions for further strengthening of embedding results.

Abstract

Suppose we wish to embed an (associative) $k$-algebra $A$ in a $k$-algebra $R$ generated in some specified way; e.g., by two elements, or by copies of given $k$-algebras $A_1,$ $A_2,$ $A_3.$ Several authors have obtained sufficient conditions for such embeddings to exist. We prove here some further results on this theme. In particular, we merge the ideas of existing constructions based on two generating <i>elements</i>, and on three given <i>subalgebra</i>, to get a construction using two given subalgebras. We pose some questions on how these results can be further strengthened.