nLab Ring

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.

Related categories

• Mod

• CRing

• Rng

• Rig

• Mon, CMon

References

• Stacks Project, 10.106 Epimorphisms of rings

category: category