An analogue of Rogers' theorem on sieving in commutative rings

TL;DR

This paper characterizes when Rogers' theorem on sieving applies in commutative rings, showing it holds precisely for Dedekind domains and certain finite rings.

Contribution

It establishes exact conditions under which Rogers' sieving theorem analogues are valid in various classes of commutative rings.

Findings

Sieving analogue holds iff the ring is a Dedekind domain.

For finite rings, the analogue holds iff the ring is a product of local rings with linearly ordered ideals.

Provides a complete characterization of rings satisfying the sieving property.

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.