Single elements. (Q2488613)

In a ring or nearring, an element \(x\) is said to be single if \(axb=0\) implies \(ax=0\) or \(xb=0\). A ring is called a generalized domain if all of its elements are single. The class of all generalized domains is closed under subrings and ultraproducts, but not under factor rings, as an example shows. In this article, it is shown that a \(0\)-simple ring with a primitive idempotent is a generalized domain. Here \(0\)-simplicity means that for all elements \(x\) and \(y\), with \(y\not=0\), there are elements \(a\) and \(b\) such that \(x=ayb\). \(0\)-simple rings are necessarily simple, and examples of \(0\)-simple rings include division rings and purely infinite \(C^*\)-algebras. Some properties of single elements in nearrings are also considered. In particular, if \(N=M_A^0(G)\) is a centralizer nearring where \(A\not=\{\text{id}\}\) is a group of automorphisms of \(G\), then every minimal left ideal, if exists, consists of single elements. Moreover, if \(N\) is regular, then minimal \(N\)-subgroups exist and consist of single elements, and every \(N\)-subgroup contains a minimal \(N\)-subgroup.