Some small-centralizer properties for rings. (Q846755)

The authors study rings \(R\) with certain finiteness conditions on certain centralizers \(C_R(x)\) of elements \(x\in R\setminus Z(R)\), where \(Z(R)\) is the center of \(R\). The main results are the following: Theorem 2.1. If \(R\) is a ring such that \(C_R(x)\) is finite for all \(x\in R\setminus Z(R)\), then \(R\) is either finite or commutative. Let \(D(R)\) denote the set of zero divisors of a ring \(R\). Theorem 2.2. Let \(R\) be a ring such that \(C_R(x)\cap D(R)\) is finite for all \(x\in D(R)\setminus Z(R)\). Then either \(R\) is finite or \(D(R)\subseteq Z(R)\). An element \(x\) of a ring \(R\) is called extremely noncommutative if \(C_R(x)=x\mathbb{Z}[x]\) -- i.e., if \(C_R(x)\) is the subring generated by \(x\). Roughly speaking, an extremely noncommutative element is one which only commutes with the ``obvious'' elements. Theorem 3.1. If \(R\) is a ring in which all noncentral elements are extremely noncommutative, then \(R\) is either finite or commutative. Theorem 3.3. Let \(R\) be a ring in which every element of \(D(R)\setminus Z(R)\) is extremely noncommutative. Then either \(R\) is finite or \(D(R)\subseteq Z(R)\). Theorem 3.9. Let \(R\) be a finite ring with \(D(R)\nsubseteq Z(R)\) such that each \(d\in D(R)\setminus Z(R)\) is extremely noncommutative. Then either \(R\) is isomorphic to a matrix ring of the form \(\text{GF}(p)e_{11}+\text{GF}(p)e_{12}\) or \(\text{GF}(p)e_{11}+\text{GF}(p)e_{21}\), or \(R\) is nil.