Identities of sums of commutative subalgebras
\textit{N. Itô} [Math. Z. 62, 400-401 (1955; Zbl 0064.25203)] proved that a group which is a product of two abelian subgroups is metabelian, i.e. 2-step solvable, and this result has stimulated investigations in similar spirit for Lie and associative algebras. In the paper under review the authors give a short and straightforward proof that if a ring \(R\) is a sum of two commutative subrings then \(R\) satisfies the polynomial identity \([x,y][z,t]=0\). They also establish that this identity is minimal, i.e. determines the variety of associative rings generated by all rings presented as sums of two commutative subrings. Recently [\textit{Yu. Bakhturin} and \textit{O. H. Kegel}, Commun. Algebra 23, 2975-2990 (1995; Zbl 0837.16020)] these results were generalized for superalgebras.
scientific article; zbMATH DE number 760329