Some embedding results for associative algebras

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.