*When Two Definitions of an Additive Functor of Commutative Algebras*

*Agree*

**David E. Dobbs **

Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA

Pages 38-69 | Received 20 October 2022, Accepted 29 December 2022, Published 16 June 2023

**Abstract**

Let R be a commutative ring and \underline{C}_R the category of commutative unital R-algebras. We show that \underline{C}_R is a pre-additive category if and only if R is a zero ring. When these conditions hold, a functor F from \underline{C}_R to a pre-additive category \underline{D} with finite products is an additive functor (in the classical sense) if and only if F is additive in the sense due to Chase-Harrison-Rosenberg (the latter sense of “additive functor” meaning that F commutes with finite products), if and only if F(R) is a terminal object of \underline{D}. More generally, if \underline{C} and \underline{D} are additive categories (that is, pre-additive categories with finite products) and F:\underline{C} \to \underline{D} is a functor, then F is additive if and only if F commutes with finite products. For such categories \underline{C} and \underline{D}, we also give four other new characterizations of the additive functors F:\underline{C} \to \underline{D}.

**Keywords**: Commutative ring, unital algebra, pre-additive category, additive functor, CHR-additive functor, zero ring, sheaf.

**MSC numbers**: Primary 13A99; Secondary 18E05.

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