The Norm Function for Commutative <semantics>Z2</semantics>-Graded Rings - Azzh Saad Alshehry; Rashid Abu-Dawwas (Axioms)
Consider a commutative <semantics>Z2</semantics>-graded ring (<semantics>R=R0⨁R1</semantics>). Consequently, each element (<semantics>x∈R</semantics>) can be uniquely expressed as <semantics>x=x0+x1</semantics>, where <semantics>x0∈R0</semantics> and <semantics>x1∈R1</semantics>. For any <semantics>x∈R</semantics>, we consider the function <semantics>N(x)=<msubsup>x02</msubsup>−<msubsup>x12</msubsup></semantics>. 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).