# nLab Ring

The category or rings

### Context

#### Algebra

higher algebra

universal algebra

category theory

# The category $Ring$ or rings

## Definition

Ring is the category of rings (with unit) and ring homomorphisms (that preserve the unit).

A ring is a monoid in Ab, where $Ab$ is the category of abelian groups. So, $Ring$ is an example of a category of internal monoids.

## Properties

### Epi/Monomorphisms

Every surjective homomorphism of rings is an epimorphism in $Ring$, but not every epimorphism is surjective.

A counterexample is the defining inclusion $\mathbb{Z} \hookrightarrow \mathbb{Q}$ of the ring of integers into the ring of rational numbers. This is an injective epimorphism of rings.

For more see for instance at Stacks Project, 10.106 Epimorphisms of rings.

## References

category: category

Last revised on May 31, 2017 at 05:42:51. See the history of this page for a list of all contributions to it.