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An analogue of Rogers' theorem on sieving in commutative rings

Petr Kucheriaviy

Abstract

We prove that an analogue of Rogers' theorem on sieving holds for an order if and only if the order is a Dedekind domain. We also prove that it holds for a finite commutative ring if and only if the ring is a direct product of local rings with linearly ordered ideals.

An analogue of Rogers' theorem on sieving in commutative rings

Abstract

We prove that an analogue of Rogers' theorem on sieving holds for an order if and only if the order is a Dedekind domain. We also prove that it holds for a finite commutative ring if and only if the ring is a direct product of local rings with linearly ordered ideals.
Paper Structure (3 sections, 8 theorems, 6 equations)

This paper contains 3 sections, 8 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Rogers for finite rings']}
  3. Proof of Theorem \ref{['Rogers for orders']}

Key Result

Theorem 1

Let $H_1, \ldots, H_r$ be arithmetic progressions in the ring of integers $\mathbb{Z}$. Then for any integers $a_1, \ldots, a_r$. Here $d$ denotes the arithmetic density.

Theorems & Definitions (17)

  • Theorem : Rogers' theorem
  • Definition 1
  • Proposition 1
  • proof
  • Example 1
  • Proposition 2
  • proof
  • Theorem 1
  • Example 2
  • Theorem
  • ...and 7 more