N-r-Ideals of Commutative Z_2-Graded Ring

Abstract

Let $R$ be a commutative ring with nonzero unity $1$. This article introduces and investigates new classes of ideals in \(\mathbb{Z}_2\)-graded rings, building on the previously established notion of \(r\)-ideals. Using the function \(N: R \to R_0\), defined by \(N(x) = x_0^2 - x_1^2\) for \(x = x_0 + x_1 \in R\), we define and study \(N\)-$r$ ideals and semi \(N\)-$r$-ideals. A proper ideal \(I\) is \(N\)-$r$-ideal if \(xy\in I\) implies \(N(x) \in I\) or $y\in zd(R)$, while it is semi \(N\)-$r$-ideal if \(x^2 \in I\) implies \(N(x) \in I\) or $x\in zd(R)$, where $zd(R)$ is the set of zero divisors of $R$. Fundamental properties of these ideals are explored, including their relationships to existing structures in graded ring theory. These results extend the understanding of ideal theory in the context of \(\mathbb{Z}_2\)-graded rings and offer new perspectives for future research.