On subrings of free rings
Let k be a commutative associative domain with unity, and M one of the following varieties: (i) variety of all k-algebras; (ii) variety of all (anti) commutative algebras; (iii) variety of all Lie algebras; (iv) variety of all Lie p-algebras, \(p=char k.\) Let F be a free M-algebra. A subalgebra B in F is called isolated if ax\(\in B\), where \(a\in k\setminus 0\), implies \(x\in B\). A subalgebra B in F is diagonal if there exists a free generating set X in F and elements \(a_ x\in k\setminus 0\) such that B is generated by \(a_ xx\), \(x\in X\). If C is a subalgebra in F and k a principal ideal ring, then there exists a diagonal subalgebra D in F such that \(D\subseteq C\subseteq F\). For some special domains k it is proved the existence of algorithms deciding whether (i) a system of equations over F is soluble; (ii) for two elements f,g\(\in F\) there exists an endomorphism a in F such that \(a(f)=g\).
scientific article; zbMATH DE number 4191933