nLab Ring

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

Revised on May 31, 2017 05:42:51 by Urs Schreiber (131.220.184.222)