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On Nilcompactifications of Prime Spectra of Commutative Rings

Lorenzo Acosta G., I. Marcela Rubio P.

Abstract

Given a ring R and S one of its proper ideals, we obtain a compactification of the prime spectrum of S through a mainly algebraic process. We name it the R-nilcompactification of SpecS. We study some categorical properties of this construction.

On Nilcompactifications of Prime Spectra of Commutative Rings

Abstract

Given a ring R and S one of its proper ideals, we obtain a compactification of the prime spectrum of S through a mainly algebraic process. We name it the R-nilcompactification of SpecS. We study some categorical properties of this construction.

Paper Structure

This paper contains 7 sections, 23 theorems, 12 equations, 7 figures.

Table of Contents

  1. Introduction.
  2. The mechanism of nilcompactification
  3. Functorial behavior of the mechanism of nilcompactification
  4. First variation
  5. Second variation
  6. Third variation
  7. Two examples

Key Result

Lemma 1

The function $\psi :\mathcal{J}(S)\rightarrow \mathcal{J}(R)$ has the following properties: (i) If $P$ is a prime ideal of $S$ then $\psi (P)$ is a prime ideal of $R$ not containing $S.$ (ii) If $P$ and $Q$ are prime ideals of $S$ such that $\psi (P)=\psi (Q)$ then $P=Q.$ (iii) If $Q$ is a prime ide

Figures (7)

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  • ...and 2 more figures

Theorems & Definitions (40)

  • Lemma 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Remark 4
  • Corollary 5
  • Definition 6
  • Theorem 7
  • proof
  • ...and 30 more