Ring

A ring is a Monoid Object in the Category of Abelian Groups with respect to the Tensor Product of Abelian Groups.

Less concisely, a ring is an H-Set $R$ equipped wht a pair of Binary Operations $+,\cdot:R\times R\to R$ and a pair of constants $0,1:R$ such that:

• $(R,+,0)$ is an Abelian Group
• $(A,\cdot,1)$ is a Monoid
• $\cdot$ is Bilinear Morphism of Monoids

As decategorifiations of abelian categories

A ring can be viewed as a single object Abelian Category.