The Norm Function for Commutative Z2-Graded Rings

Consider a commutative Z2-graded ring (R=R0⨁R1). Consequently, each element (x∈R) can be uniquely expressed as x=x0+x1, where x0∈R0 and x1∈R1. For any x∈R, we consider the function N(x)=x02−x12. In this work, we examine the properties of N and utilize them to derive new results. Moreover, we apply this function to establish concepts such as N-prime ideals, N radicals, N-integral domains, and N fields, achieving several notable results along the way. Among our results, we demonstrate that an N-prime ideal is not necessarily prime. Additionally, we show that the N radical differs from the usual radical ideal and is not always an ideal. Furthermore, we establish that an N-integral domain (N field) is not necessarily an integral domain (field).