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

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.