On a conjecture by Herstein
Let P be the set of positive integers. Let R be an arbitrary ring, with N(R), \(N_ r(R)\), and Z(R) denoting respectively the nil radical, the sum of all nil right ideals, and the center. For elements \(x_ 1,x_ 2,..\). of R define extended commutators \([x_ 1,x_ 2,...,x_{k+1}]\) by \([x_ 1,x_ 2]=x_ 1x_ 2-x_ 2x_ 1\) and \([x_ 1,...,x_{n+1}]=[[x_ 1,...,x_ n],x_{n+1}]\); and if \(x_ 1=x\) and \(x_ 2=x_ 3=...=x_{k+1}=y\), abbreviate the notation to \([x,y]_ k\). For \(k\geq 1\), define \(T_ k(R)\) as the set of \(x\in R\) such that for each \(y\in R\), there exists \(n=n(x,y)\in P\) for which \([x,y^ n]_ k=0\); call R a \(C_ k\)-ring if for each \(x,y\in R\), P contains \(m=m(x,y)\) and \(n=n(x,y)\) for which \([x^ m,y^ n]_ k=0\). Extend these definitions by letting k vary with x and y, thereby obtaining the set T(R) and the notion of C-ring. The hypercenter theorem of \textit{I. N. Herstein} [J. Algebra 36, 151-157 (1975; Zbl 0313.16036)] asserts that if \(N(R)=\{0\}\), then \(T_ 1(R)=Z(R)\); and a subsequent result of Herstein [J. Algebra 38, 112-118 (1976; Zbl 0323.16014)] is that every \(C_ 1\)-ring with \(N(R)=\{0\}\) is commutative. It has long been conjectured that the same conclusions hold for \(T_ k(R)\) and \(C_ k\)-rings, and various special cases have been established. The present paper proves the conjectures in general - specifically, that (1) for each \(k\in P\), if \(N(R)=\{0\}\), then \(T_ k(R)=Z(R)\); and (2) for arbitrary \(k\in P\), every \(C_ k\)-ring with \(N(R)=\{0\}\) is commutative. Also included are proofs of the following: (3) if \(N_ r(R)=\{0\}\), then \(T(R)=Z(R)\); (4) if R is a C-ring with \(N_ r(R)=\{0\}\), then R is commutative. It is natural to conjecture that every C-ring with \(N(R)=\{0\}\) is commutative, and the authors conclude with a proof of equivalence of this conjecture with two other conjectures. This is an important paper. It is carefully argued and, given the difficulty of the problem, surprisingly easy to read.
scientific article; zbMATH DE number 4127453
scientific article