On algebraic embeddings of rings into groups (Q1093730)

A special verbal embedding (s.v.e) of a ring \(R\) into a group \(G\) is an injective map \(f\) from \(R\) into \(G\) such that \(f(r+s)=f(r)f(s)\) and \(f(rs)=p(f(r),f(s))\) where \(p(x,y)\) is a polynomial in variables \(x,y\) and constants from \(G\). The following results are proven: Theorem: Let \(R\) be a ring with identity and f a s.v.e. of \(R\) into a group \(G\). Suppose that \(p(x,y)=[x^ a,y^ b]\) where \(a^ 2=b^ 2=(ab)^ 3=e\). Then the normal closure in \(G\) of \(f(R)\) is an image ofthe Steinberg group \(ST(3,R)\). - Theorem: There does not exist a s.v.e. ofthe ring of integers into \(GL_ 2\) or \(PSL_ 2\) over the complex numbers. Results are also proven in the setting of embedding more general algebraic structures into groups. Among them is: Theorem: Let \(S\) be the product semigroup \((0*0=1*0=0*1=0\), \(1*1=1)\) and suppose that there is a verbal embedding \(f\) of \(S\) into the group \(G\). Then the normal closure of \(f(S)\) in \(G\) is not residually solvable. An immediate consequence is: Corollary: If a ring with identity is verbally embedded in a group \(G\) then \(G\) is not residually solvable.