Quasinilpotents in rings and their applications

An element a of an associative ring R is said to be quasinilpotent if 1 − ax is invertible for every x ∈ R with xa = ax. Nilpotents and elements in the Jacobson radical of a ring are well-known examples of quasinilpotents. In this paper, properties and examples of quasinilpotents in a ring are provided, and the set of quasinilpotents is applied to characterize rings with some certain properties.