Noncommutativity and noncentral zero divisors
For \(R\) any associative ring, let \(Z\) denote the center of \(R\) and \(D\) the set of all zero divisors in \(R\). The authors prove that if \(1\in R\) is a finite noncommutative ring, \(R\cong S_1\oplus\cdots\oplus S_k\) for indecomposable rings \(S_j\), and \(n\geq 1\) is the number of noncommutative \(S_j\) then \((4^n-1)| D\cap Z|\leq| D-Z|\). When \(D\) is replaced with the set \(N\) of all nilpotent elements in \(R\) the conclusion is that \((p^n-1)| N\cap Z|\leq| N-Z|\) where \(p\) is the smallest prime divisor of \(| R|\). If \(S\) is the set of elements in \(R\) with a nonzero two sided annihilator, then for \(R\) infinite, \(| S-Z|=| D-Z|\) and \(| S-Z|\) is infinite or empty. Other similar results in the paper involve periodic, weakly periodic, or quasiperiodic rings.
scientific article; zbMATH DE number 1275558