MacHale, D. Rings that are nearly Boolean. III. (English) Zbl 0604.16028 Proc. R. Ir. Acad., Sect. A 86, 31-33 (1986). [Part II, cf. ibid. 83, 165-167 (1983; Zbl 0544.16015.] The author studies the following conjecture, a generalization of a theorem of Herstein: If R is a ring such that \(x^ n-\alpha (x)\) is central for all \(x\in R\), where n is a fixed integer \(>1\) and \(\alpha\) is an additive homomorphism of R onto R, then R is commutative. The conjecture is verified in the case \(n=2^ i+2^ j\), \(i>j>0\), and R has unity. Reviewer: M.Abad Cited in 1 Document MSC: 16U70 Center, normalizer (invariant elements) (associative rings and algebras) 06E20 Ring-theoretic properties of Boolean algebras 16U99 Conditions on elements 16E50 von Neumann regular rings and generalizations (associative algebraic aspects) Keywords:central elements; additive homomorphism; commutative Citations:Zbl 0544.16015 PDFBibTeX XMLCite \textit{D. MacHale}, Proc. R. Ir. Acad., Sect. A 86, 31--33 (1986; Zbl 0604.16028)