nLab Q-algebra

Contents

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

A ring RR is a \mathbb{Q}-algebra if there is a function j: +Rj:\mathbb{Z}_+ \to R such that for all positive integers a +a\in\mathbb{Z}_+ and elements bRb \in R, h(i(a))j(a)=1h(i(a)) \cdot j(a) = 1 and j(a)b=bj(a)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 RR has a ring homomorphism h:Rh:\mathbb{Q}\to R, which corresponds to the definition of \mathbb{Q}-algebra in terms of ring homomorphisms.

Examples

See also

Last revised on June 17, 2022 at 16:42:52. See the history of this page for a list of all contributions to it.