ON QUASI-COMMUTATIVE RINGS

Abstract

We study the structure of central elements in relation with polynomial rings and introduce quasi-commutative as a generalization of commutative rings. The Jacobson radical of the polynomial ring over a quasi-commutative ring is shown to coincide with the set of all nilpotent polynomials; and locally finite quasi-commutative rings are shown to be commutative. We also provide several sorts of examples by showing the relations between quasi-commutative rings and other ring properties which have roles in ring theory. We examine next various sorts of ring extensions of quasi-commutative rings.