# nLab Q-algebra

Contents

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

Similar to how $\mathbb{Q}$-vector spaces could be defined without using the rational numbers, as a torsion-free divisible group, there is a definition of an associative unital $\mathbb{Q}$-algebra without using the rational numbers.

## Definition

By the universal property of the ring of integers, every ring $R$ has a ring homomorphism $h:\mathbb{Z} \to R$ from the integers to $R$ which lands in the center of $R$, and there is an injection $i:\mathbb{Z}_+ \to \mathbb{Z}$ from the positive integers to the integers.

A ring $R$ is a $\mathbb{Q}$-algebra if there is a function $j:\mathbb{Z}_+ \to R$ such that for all positive integers $a\in\mathbb{Z}_+$ and elements $b \in R$, $h(i(a)) \cdot j(a) = 1$ and $j(a) \cdot b = b \cdot j(a)$.

The rational numbers $\mathbb{Q}$ are the initial $\mathbb{Q}$-algebra. As a result, every $\mathbb{Q}$-algebra $R$ has a ring homomorphism $h:\mathbb{Q}\to R$, which corresponds to the definition of $\mathbb{Q}$-algebra in terms of ring homomorphisms.

## Examples

• Every ordered field is a $\mathbb{Q}$-algebra.