Mal'cev nilpotent algebras
The algebras under study are torsion free associative algebras over a commutative domain of prime characteristic \(p\) or of characteristic \(0\). Mal'cev introduced semigroup identities that hold in groups iff the groups are nilpotent. Given \(x\), \(y\), \(w_n\), \(n\geq 2\), define \(x_1=xy\), \(y_1=yx\), \(x_{n+1}=x_nw_{n+1}y_n\), \(y_{n+1}=y_nw_{n+1}x_n\). A semigroup is called Mal'cev nilpotent of class \(n\) if it satisfies the identity \(x_n=y_n\) and does not satisfy \(x_{n-1}=y_{n-1}\). If \(A\) is a ring, the circle semigroup of \(A\) is denoted by \((A;\circ)\) where \(x\circ y=x+y+xy\). Theorem 1: An algebra \(A\) is Lie nilpotent of class \(n\) iff \((A;\circ)\) is Mal'cev nilpotent of class \(n\). Theorem 2: If \(R\) is a ring that is Lie nilpotent of class \(n\), then the multiplicative semigroup and the circle semigroup of \(R\) are Mal'cev nilpotent of class at most \(n\). This generalizes a result of \textit{N. D. Gupta} and \textit{F. Levin} [J. Algebra 81, 225-231 (1983; Zbl 0514.16024)]. Theorem 3: Let \(A\) be an algebra with \(1\) that is generated by its nilpotent elements. Then its group of units is nilpotent of class at most \(n\) iff \(A\) is Lie nilpotent of class at most \(n\).
scientific article; zbMATH DE number 1292443