An analogue for rings of a group problem of P. Erdős and B. H. Neumann
In response to a question posed by Erdős, B. H. Neumann studied groups having no infinite subsets of pairwise non-commuting elements, which he characterized as those with center of finite index. Following Neumann's terminology, the authors define a ring \(R\) to be a Paul Erdős ring, or PE-ring, if every set of pairwise non-commuting elements is finite. They show that the center of every PE-ring is of finite index and establish several equivalent characterizations of PE-rings. They also discuss near-commutativity of PE-rings, in several different senses of near-commutativity and characterize rings with finite commutator ideal.
scientific article; zbMATH DE number 1225562