The homogeneous spectrum of a $\Bbb Z_2$-graded commutative ring

TL;DR

This paper explores the structure of $Z_2$-graded commutative rings, establishing a homeomorphism between their graded spectrum and the spectrum of the degree-zero part, linking graded ideals to classical prime ideals.

Contribution

It demonstrates a strong correspondence between $Z_2$-graded prime/maximal ideals and those of the degree-zero component, revealing topological equivalence of their spectra.

Findings

The $Z_2$-graded spectrum of $R$ is homeomorphic to the spectrum of $R_0$.

Prime and maximal ideals in the graded setting correspond to those in the degree-zero part.

The topological structure of the spectrum is preserved under this correspondence.

Abstract

Let $\Bbb Z_2:=\Bbb Z/2\Bbb Z$ be the additive group with two elements. In this article, we focus only on $\Bbb Z_2$-graded commutative ring i.e commutative ring $R$ such that $R=R_0\oplus R_1$ as Abelian group and $R_iR_j\subseteq R_{i+j}$ for all $i,j\in \Bbb Z_2$. Our main goals is to establish a strong relation between $\Bbb Z_2$-graded prime ( maximal ) ideals of $R$ and prime ( maximal) ideals of $R_0$, for instance, it is showed that, the $\Bbb Z_2$-graded spectrum of $R$ is homeomorphic to the spectrum of $R_0$ with respect to the Zariski topologies.