Differential commutator identities.
The main motivation for this work is a result and a question of \textit{I. N. Herstein} [Can. Math. Bull. 21, 369-370 (1978; Zbl 0412.16018)] which show how differential identities arise. He shows that if \(d\) is a nonzero derivation of a prime ring \(R\), and if \(R^d\), the image of \(R\) under \(d\), is commutative, then \(R\) is commutative unless both \(\text{char\,}R=2\) and \(R\) satisfies the standard identity \(S_4\): \(R\) embeds in some \(M_2(F)\). The author considers a natural generalization of this result for the class of polynomials \(E_n(X)=[E_{n-1}(x_1,\dots,x_{n-1}),x_n]\). Employing sophisticated matrix computations, he proves that if \(R\) satisfies a differential identity \(E_n(x_1^{d_1},\dots,x_n^{d_n})\), or \([E_n(\dots,x_i^{d_i},\dots),E_m(\dots,y_j^{h_j},\dots)]\) with some restrictions, then \(R\) must embed in \(M_2(F)\), but that differential identities using \([[E_n,E_m],E_s]\) with \(m,n,s>1\) need not force \(R\) to embed in \(M_2(F)\). These results hold if the expressions are identities for a noncommutative Lie ideal of \(R\), rather than for \(R\) itself.
scientific article; zbMATH DE number 5775703
scientific article