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Some remarks on Prüfer rings with zero-divisors

Federico Campanini, Carmelo Antonio Finocchiaro

Abstract

Let $A$ be the fiber product $R\times_TB$, where $B\to T$ is a surjective ring homomorphism with regular kernel and $R\subseteq T$ is a ring extension where $T$ is an overring of $R$. In this paper we provide a characterization of when $A$ has distinguished Prüfer-like properties and new constructions of Prüfer rings with zero-divisors. Furthermore we give examples of homomorphic images of Prüfer rings that are Prüfer without assuming that the kernel of the surjection is regular. Finally we provide some remarks on the ideal theory of pre-Prüfer rings.

Some remarks on Prüfer rings with zero-divisors

Abstract

Let be the fiber product , where is a surjective ring homomorphism with regular kernel and is a ring extension where is an overring of . In this paper we provide a characterization of when has distinguished Prüfer-like properties and new constructions of Prüfer rings with zero-divisors. Furthermore we give examples of homomorphic images of Prüfer rings that are Prüfer without assuming that the kernel of the surjection is regular. Finally we provide some remarks on the ideal theory of pre-Prüfer rings.
Paper Structure (6 sections, 25 theorems, 5 equations)

This paper contains 6 sections, 25 theorems, 5 equations.

Table of Contents

  1. Notation and preliminaries
  2. Prüfer extensions
  3. Prüfer conditions in distinguished pullbacks
  4. Homomorphic images and Prüfer rings
  5. Regular morphisms
  6. Pre-Prüfer rings

Key Result

Theorem 1.2

Gr A ring $R$ is a Prüfer ring if and only if the pair $(R,\mathfrak{m})$ has the regular total order property, for every maximal ideal of $R$.

Theorems & Definitions (52)

  • Definition 1
  • Remark 1.1
  • Theorem 1.2
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • ...and 42 more