NJ-Abelian Rings: an Abelian-Like Approach

This article extends the concept of NJ-semicommutative rings to introduce the broader class of NJ-abelian rings, which are defined by properties involving nilpotent elements and the Jacobson radical. We investigate the unique algebraic properties of NJ-abelian rings and analyze their relationships with various types of rings, including abelian, reduced, J-clean, local, and Dedekind-finite rings. In particular, we show that every NJ-semicommutative ring is NJ-abelian, much like every semicommutative ring is abelian.