Heredity of radicals in a class of noncommutative Jordan rings
A ring is called noncommutative Jordan if it satisfies the identities \((x, y, x)= (x^2, y, x)= 0\). Let \(R\) be an associative commutative ring containing \(1/6\) and \({\mathcal M}\) the class of noncommutative Jordan rings over \(R\) satisfying the identity \(([x, y], z,z) =0\). The author proves that if \(r\) is a radical in \({\mathcal M}\) then an ideal of an \(r\)- semisimple ring is \(r\)-semisimple. This implies that the locally nilpotent radical in \({\mathcal M}\) is hereditary. (The existence of the locally nilpotent radical in \({\mathcal M}\) has been proved by \textit{I. P. Shestakov} [Algebra Logic 10, 252-280 (1973); translation from Algebra Logika 10, 407-448 (1971; Zbl 0259.17001)]).
scientific article; zbMATH DE number 795656