On commutators and derivations in rings.
Let \(a\) be a fixed element of the ring \(R\); and for each \(x_0\in R\), define higher commutators \(x_1,x_2,\dots\) inductively by \(x_i=[a,x_{i-1}]\). The authors' main purpose is to study when products \(b_ic_j\) or integer multiples of such products lie in the ideal generated by some power of \(a\). The principal theorem, for arbitrary rings, is rather technical; but one easily-stated consequence is the following: Let \(A\) be an algebra over a field \(F\) of characteristic \(0\), and let \(a,b_0,c_0\in A\) such that \([b_i, c_j]=0\) for all \(i,j\geq 0\). If \(s\) and \(t\) are odd positive integers and \(n=(s+t+2)/2\), then \(b_sc_t\) is in the ideal of \(A\) generated by \(a^n\). The final section of the paper gives applications to the ring of additive maps of a ring \(A\) into itself, with the roles of \(a,b_0,c_0\) being played respectively by a derivation, a left multiplication map \(L_x\) and a right multiplication map \(R_y\).
scientific article; zbMATH DE number 2105352
scientific article