nLab
ring object

Idea

For $C$ a cartesian monoidal category (a category with finite products), an internal ring or a ring object in $C$ is an internalization to the category $C$ of the notion of a ring.

This is a monoid object internal to the category of abelian group objects internal to $C$.

Ring objects can be defined in more general symmetric monoidal categories as the corresponding module over a ring operad.

Examples

• A ring object in Top is a topological ring.

• A topos equipped with a ring object is called a ringed topos, see there for more details.

Related concepts

• monoid, monoid object,

• group, group object, group object in an (∞,1)-category

• ring, ring object