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

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