linguistic:paraphrasing, structural:scholar_edition
Ring object in a category

Let C be a category with finite products. Let pt denote a terminal object of C (an empty product). A ring object in C is an object R equipped with morphisms R × R → a R {\displaystyle R\times R\;{\stackrel {a}{\to }}\,R} (addition), R × R → m R {\displaystyle R\times R\;{\stackrel {m}{\to }}\,R} (multiplication), pt → 0 R {\displaystyle \operatorname {pt} {\stackrel {0}{\to }}\,R} (additive identity), R → i R {\displaystyle R\;{\stackrel {i}{\to }}\,R} (additive inverse), and pt → 1 R {\displaystyle \operatorname {pt} {\stackrel {1}{\to }}\,R} (multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an object R equipped with a factorization of its functor of points h R = Hom (−, R): C op → S e t s {\displaystyle h_{R}=\operatorname {Hom} (-,R):C^{\operatorname {op} }\to \mathbf {Sets} } through the category of rings: C op → R i n g s ⟶ forgetful S e t s. {\displaystyle C^{\operatorname {op} }\to \mathbf {Rings} {\stackrel {\textrm {forgetful}}{\longrightarrow }}\mathbf {Sets}.}