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 = a b a . 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 p n . 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.