# Non-unital algebra objects of stable symmetric monoidal model categories by Smith ideal theory

## TL;DR

This paper explores the relationship between non-unital and augmented unital algebra objects within stable symmetric monoidal model categories, utilizing Smith ideal theory to generalize their correspondence.

## Contribution

It introduces a framework connecting non-unital algebra objects to augmented unital algebras via Smith ideal theory in stable symmetric monoidal model categories.

## Key findings

- Established a generalized correspondence between non-unital and augmented algebra objects.
- Applied Smith ideal theory to stable symmetric monoidal model categories.
- Provided a new perspective on algebra object structures in homotopical algebra.

## Abstract

This note remarks that the correspondence between non-unital algebras and augmented unital algebras can be derived from Hovey's Smith ideal theory. Applying Smith ideal theory of stable symmetric monoidal model category, we formulate non-unital algebra objects of stable symmetric monoidal model categories and generalize the correspondence between non-unital algebra objects and augmented algebra objects.