Journal of the
Korean Mathematical Society JKMS

This paper is intended as a discussion of some generalizations of the notion of a reversible ring, which may be obtained by the restriction of the zero commutative property from the whole ring to some of its subsets. By the INCZ property we will mean the commutativity of idempotent elements of a ring with its nilpotent elements at zero, and by ICZ property we will mean the commutativity of idempotent elements of a ring at zero. We will prove that the INCZ property is equivalent to the abelianity even for nonunital rings. Thus the INCZ property implies the ICZ property. Under the assumption on the existence of unit, also the ICZ property implies the INCZ property. As we will see, in the case of nonunital rings, there are a few classes of rings separating the class of INCZ rings from the class of ICZ rings. We will prove that the classes of rings, that will be discussed in this note, are closed under extending to the rings of polynomials and formal power series.

Keywords: idempotents, nilpotents, symmetric rings, reversible rings, abelian rings

MSC numbers: Primary 16U80, 16U99

Supported by: This work was financially supported by MNiSW under subsidy for maintaining the research potential of WMiI UwB (the ?rst author) and the research theme No. 493/17/S (the second author).