[Paper Review] A Note on Primary Spectrum Over Commutative Rings

Neslihan Ayşen Özkirişçi, Zeliha Kılıç+1 more|arXiv (Cornell University)|May 22, 2017

Rings, Modules, and Algebras2 references5 citations

TL;DR

This paper introduces a new Zariski topology on the set of primary ideals of a commutative ring R, termed the primary spectrum $\mathrm{Prim}(R)$, by defining closed sets via $V_{\mathrm{rad}}(I) = \{ Q \in \mathrm{Prim}(R) \mid I \subseteq \sqrt{Q} \}$. The key contribution is establishing that this construction yields a well-defined topological space with distinct properties from the classical prime spectrum, particularly in zero-dimensional and chain rings, where $V_{\mathrm{rad}}(I)$ is often the entire space, unlike the prime spectrum where it is finite or small.

ABSTRACT

In this work we define a primary spectrum of a commutative ring R with its Zariski topology $\mathfrak{T}$. We introduce several properties and examine some topological features of this concept. We also investigate differences between the prime spectrum and our primary spectrum.

Motivation & Objective

• To define a new topological structure on the set of primary ideals of a commutative ring, generalizing the prime spectrum beyond principal ideal domains.
• To investigate how the primary spectrum differs topologically from the classical prime spectrum, especially in rings with non-trivial nilpotents or finite chains of ideals.
• To analyze the Zariski topology on $\mathrm{Prim}(R)$, defined via the variety $V_{\mathrm{rad}}(I) = \{ Q \in \mathrm{Prim}(R) \mid I \subseteq \sqrt{Q} \}$, and establish its axioms.
• To characterize conditions under which $\mathrm{Prim}(R)$ satisfies separation axioms ($T_0$, $T_1$, $T_2$), linking them to ring-theoretic properties like being a $P$-ring.
• To explore applications of the primary spectrum in zero-dimensional rings and chain rings, showing that $V_{\mathrm{rad}}(I)$ is often the entire space, contrasting with the prime spectrum.

Proposed method

• Define the primary variety $V_{\mathrm{rad}}(I)$ as the set of all primary ideals $Q$ such that $I \subseteq \sqrt{Q}$, for any ideal $I$ of $R$.
• Prove that the family $\{V_{\mathrm{rad}}(I) \mid I \trianglelefteq R\}$ satisfies the axioms of closed sets for a topology on $\mathrm{Prim}(R)$, called the Zariski topology.
• Establish that $V_{\mathrm{rad}}(I) = V_{\mathrm{rad}}(\sqrt{I})$, showing the topology depends only on the radical of the ideal.
• Use the closure operator: $\mathrm{Cl}(\{I\}) = V_{\mathrm{rad}}(I)$, and show that $\mathrm{Cl}(Y) = V_{\mathrm{rad}}(\xi(Y))$ for any subset $Y \subseteq \mathrm{Prim}(R)$.
• Analyze separation axioms by relating $V_{\mathrm{rad}}(I) = V_{\mathrm{rad}}(J)$ to equality $I = J$, and prove that $\mathrm{Prim}(R)$ is $T_0$ iff $R$ is a $P$-ring.
• Apply the theory to zero-dimensional and chain rings (e.g., $\mathbb{Z}_{p^m}$, $k[X]/(X^n)$, $\mathrm{GR}(p^s, p^{sm})$), showing $V_{\mathrm{rad}}(I) = \mathrm{Prim}(R)$ for all proper ideals $I$.

Experimental results

Research questions

• RQ1 How does the primary spectrum $\mathrm{Prim}(R)$ with the $V_{\mathrm{rad}}$ topology differ from the classical prime spectrum $\mathrm{Spec}(R)$ in terms of topological structure and closure properties?
• RQ2 Under what ring-theoretic conditions is the primary spectrum $\mathrm{Prim}(R)$ a $T_0$, $T_1$, or $T_2$ space?
• RQ3 What is the behavior of $V_{\mathrm{rad}}(I)$ in zero-dimensional and chain rings, and how does it compare to $V(I)$ in the prime spectrum?
• RQ4 Can the closure of a singleton $\{I\}$ in $\mathrm{Prim}(R)$ be characterized, and what does this imply about the topology?
• RQ5 How does the condition $\sqrt{\bigcap_{\lambda \in \Gamma} I_\lambda} = \bigcap_{\lambda \in \Gamma} \sqrt{I_\lambda}$ relate to the embeddability of a ring into a zero-dimensional ring?

Key findings

• The family $\{V_{\mathrm{rad}}(I) \mid I \trianglelefteq R\}$ satisfies the axioms of closed sets, thus defining a well-behaved Zariski topology on $\mathrm{Prim}(R)$, making $\mathrm{Prim}(R)$ a topological space.
• In finite chain rings such as $\mathbb{Z}_{p^m}$, $k[X]/(X^n)$, and $\mathrm{GR}(p^s, p^{sm})$, $V_{\mathrm{rad}}(I) = \mathrm{Prim}(R)$ for every proper ideal $I$, while $V(I)$ is a singleton, showing a fundamental topological difference from the prime spectrum.
• The closure of a singleton $\{I\}$ in $\mathrm{Prim}(R)$ is $\mathrm{Cl}(\{I\}) = V_{\mathrm{rad}}(I)$, and $J \in \mathrm{Cl}(\{I\})$ if and only if $I \subseteq \sqrt{J}$.
• The primary spectrum $\mathrm{Prim}(R)$ is a $T_0$-space if and only if $R$ is a $P$-ring, i.e., every primary ideal is maximal.
• The primary spectrum $\mathrm{Prim}(R)$ is a $T_2$-space if and only if $R$ is a $P$-ring, and this is equivalent to it being $T_1$, showing strong topological equivalence in this case.
• For zero-dimensional rings, $\mathrm{Cl}(Y) = V_{\mathrm{rad}}(\xi(Y))$, where $\xi(Y)$ is the intersection of all ideals in $Y$, providing a closure formula analogous to the prime spectrum.