On some center-like subsets of rings
The authors obtain some interesting commutativity results involving higher commutators. Specifically, for any ring R and any x,y\(\in R\), define \([x,y]_ 0=x\), \([x,y]_ 1=xy-yx\), and for \(k>1\), \([x,y]_ k=[[x,y]_{k-1},y]\). Set \(T(R)=\{x\in R|\) for each \(y\in R\) there are n and k depending on both x and y so that \([x,y^ n]_ k=0\}\) and \(K(R)=\{x\in R|\) for some n and k depending on x, \([x,y^ n]_ k=0\) for all \(y\in R\}\). Finally, for each positive integer k let \(B_ k(R)=\{x\in T(R)|\) for each \(y\in R\) \([x,y^ n]_ k=0\}\). Clearly, each of these sets contains Z(R), the center of R, and the main results give conditions which force each to be equal to Z(R). In particular, \(T(R)=Z(R)\) when R is a semi-simple ring; \(K(R)=Z(R)\) when R is a semi- prime ring; and \(B_ k(R)=Z(R)\) when it is a subring of R, provided that R has an identity element and contains no nonzero nil ideals.
scientific article; zbMATH DE number 4008602