The homogeneous spectrum of a ${\mathbb{Z}}_2$-graded commutative ring

Let ${\mathbb{Z}}_2:=\mathbb{Z}\mathrm{/}2\mathbb{Z}$ be the additive group with two elements. In this article, we focus only on ${\mathbb{Z}}_2$-graded commutative ring i.e commutative ring $R$ such that $R=R_0\oplus R_1$ as Abelian group and $R_i{R}_j\subseteq R_{i+j}$ for all $i,j\in {\mathbb{Z}}_2$. Our main goals is to establish a strong relation between ${\mathbb{Z}}_2$-graded prime ( maximal ) ideals of $R$ and prime ( maximal) ideals of $R_0$, for instance, it is showed that, the ${\mathbb{Z}}_2$-graded spectrum of $R$ is homeomorphic to the spectrum of $R_0$ with respect to the Zariski topologies.