Jeff Smith's Theory of Ideals

TL;DR

This paper examines Jeff Smith's 2006 theory of ideals within triangulated symmetric monoidal categories, demonstrating its equivalence to a specific bimodule map, thereby clarifying its algebraic structure.

Contribution

It establishes the equivalence between Smith's ideal definition and a central bimodule map, providing a clearer algebraic understanding of the theory.

Findings

Smith's ideals are equivalent to central bimodule maps

The paper clarifies the algebraic structure of ideals in ring spectra

Provides a foundation for further algebraic exploration in monoidal categories

Abstract

In 2006, Jeff Smith proposed a theory of ideals for rings in a triangulated symmetric monoidal category such as ring spectra or DGAs. We show that his definition is equivalent to a `central' $R$-$R$-bimodule map $ I \to R$.