nLab reciprocal algebra

Idea

An algebra with a reciprocal that behaves like $\frac{1}{x}$ in the rational and real numbers.

Definition

A reciprocal algebra is a possibly non-associative algebra $A$, typically over a field $k$, which has the property that for non-zero element $b$, there exists an element $c$, called the reciprocal of $b$, such that for every element $a$

• $a (c b) = a$
• $(c b) a = a$
• $a (b c) = a$
• $(b c) a = a$

Examples

• Every Cayley-Dickson algebra is a reciprocal algebra. Examples include the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, the quaternions $\mathbb{H}$, and the octonions $\mathbb{O}$.

• The ring of rational polynomials $\mathbb{R}(X)$ is a real reciprocal algebra.

• There exists a reciprocal algebra with nonzero zero divisors.

Counter-examples

• There exists a division algebra with identity that does not have two-sided inverses for every nonzero element.

Related concepts

• division algebra
• reciprocal ring