Graphs Induced by W-Regular Rings With Application to Zpn

A ring R is called W-regular if for any non-nilpotent element a of R, there exists an element b of R such that a=aba. This ring is generalized to a regular ring and is a special type of π-regular ring; therefore, some results on rings that are regular cannot be generalized to W-regular rings, for example, the direct product, singularity, and Jacobson radical. In the ring of integers modulo n, it will be regular if n is square free, but for W-regular, we add another case where the ring is W-regular. In addition, there are some properties of a W-regular ring that do not to satisfy in π-regular. In this study, investigate various properties of these rings have been studied. This paper defines a new graph based on the set of W-regular elements, with illustrative examples provided for both commutative and noncommutative rings. A detailed description of this graph was subsequently given, specifically for the ring of integers modulo p (where p is a prime number) and for the ring of integer modulus pn. Furthermore, a comprehensive general description was presented for both cases, along with. The study concluded by finding the Hosoya Polynomial for each of these graphs.