# nLab Ring

Contents

### Context

#### Algebra

higher algebra

universal algebra

category theory

# Contents

## 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

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

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

A counterexample:

###### Proposition

In unital Rings, the canonical inclusion $\mathbb{Z} \overset{i}{\hookrightarrow} \mathbb{Q}$ of the integers into the rational numbers is an epimorphism.

###### Proof

Since every rational number is the product of an integer with the multiplicative inverse of an integer

$\frac{a}{b} \,=\, a \cdot b^{-1} \;\; \in \; \mathbb{Q}$

and since unital ring homomorphism

$\mathbb{Z} \overset{\;\; i \;\;}{\hookrightarrow} \mathbb{Q} \underoverset {\;\;g\;\;} {\;\;f\;\;} {\rightrightarrows} R \,.$

preserve multiplicative inverses, $f\left( a/b \right) = f(a) \cdot \big(f(b)\big)^{-1}$, it follows that any pair $(f,g)$ of parallel morphisms on $\mathbb{Q}$ are equal as soon as they take equal value on the integers.

category: category