nLab Q-algebra

Idea

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

See also

• rational vector space
• rational numbers
• ordered field
• associative unital algebra