On the Zariski topology over the primary-like spectrum

Hosein Fazaeli Moghimi  ; Fatemeh Rashedi

Novi Sad Journal of Mathematics, Tome 52 (2022) no. 1.

Résumé

Let $R$ be a commutative ring with identity and $M$ be a unital $R$-module. The primary-like spectrum $\mathcal{PS}(M)$ has a topology which is a generalization of the Zariski topology on the prime spectrum $\operatorname{Spec}(R)$. We get several topological properties of $\mathcal{PS}(M)$, mostly for the case when the continuous mapping $\phi:\mathcal{PS}(M)\rightarrow \operatorname{Spec}(R/{\operatorname{Ann}(M))}$ defined by $\phi(Q)=\sqrt{(Q:M)}/{\operatorname{Ann}(M)}$ is surjective or injective. For example, if $\phi$ is surjective, then $\mathcal{PS}(M)$ is a connected space if and only if $\operatorname{Spec}(R/{\operatorname{Ann}(M))}$ is a connected space. It is shown that if $\phi$ is surjective, then a subset $Y$ of $\mathcal{PS}(M)$ is irreducible if and only if $Y$ is the closure of a singleton set. It is also proved that if the image of $\phi$ is a closed subset of $ \operatorname{Spec}(R/{\operatorname{Ann}(M))}$, then $\mathcal{PS}(M)$ is a spectral space if and only if $\phi$ is injective.

Détail

Publié le : 2021-05-17

DOI : 10.30755/NSJOM.09880

Classification : 13C13, 13C99, 54B99
Keywords: sprimary-like submodule, primeful property, continuous map, irreducible space, spectral space