On Weakly 2-Invo Clean Rings With Some Properties in Graph Theory

Abstract

The concept of a weakly two-involution clean ring is presented; it is a generalization of two-involution clean ring, which allows the addition of more elements to a ring, which will be observed in its graph theoretic representation. The element in a weakly two-involution clean ring can be expressed as a sum or difference of two elements, which is an involution and an idempotent. Furthermore, if the involution and idempotent can be taken such that they commute, the ring is called a strongly weakly two-involution clean ring. We give some properties of weakly two-involution clean rings. It is also proven that R is isomorphic to all from Z3 or Z5 or Z7 when R is a weakly two-involution clean ring with 2 belonging to the set of all unit elements in R. In addition, this paper contains several results related to graph theory, including the connectedness, diameter, girth, and order of a complete subgraph of the graphs resulting from a weakly two-involution clean ring.