Varieties of associative rings with the property of embeddability of amalgams (Q793822)

Let \(\{R_{\lambda}\}_{\lambda \in \Lambda}\) be a family of rings. Let us suppose that \(R_{\lambda}\cap R_{\mu}=R_{\mu}\cap R_{\nu}=R\) and R is a subring of each ring of this family for arbitrary pairwise different \(\lambda,\mu,\nu \in \Lambda.\) Then the set-theoretic union \(\cup_{\lambda \in \Lambda}R_{\lambda}\) of the rings \(R_{\lambda}\) is called their amalgam (with the single amalgamated subring R). We will say that a variety of rings \({\mathfrak M}\) has the property of embeddability of amalgams if each amalgam of rings of \({\mathfrak M}\) can be embedded in a ring from \({\mathfrak M}.\) Theorem 1. A variety of associative rings \({\mathfrak M}\) has the property of embeddability of amalgams if and only if it is generated by a finite (possibly, empty) set of finite fields of pairwise different characteristics and a certain ring with zero multiplication.