nLab
Ring

Contents

Context

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 AbAb is the category of abelian groups. So, RingRing 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 RingRing, but not every epimorphism is surjective.

A counterexample:

Proposition

In unital Rings, the canonical inclusion i\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

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

and since unital ring homomorphism

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

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

References

category: category

Last revised on January 21, 2021 at 23:59:20. See the history of this page for a list of all contributions to it.