formally real algebra

A ring (or rig, or nonassociative ring, or indeed magma in any Ab-enriched category) is **formally real** if, whenever

$\sum_i x_i^2 = 0$

(for a finite sum), each $x_i = 0$.

One generally speaks of **formally real algebras** (possibly associative, possibly nonassociative) over the real numbers, but the concept has nothing to do with the $\mathbb{R}$-linear structure. That said, the trivial algebra is the only formally real algebra over the complex numbers; more generally, if $A \to B$ is a monomorphism of rings (or of rigs, etc), then $A$ must be formally real if $B$ is.

Of course, the real numbers themselves form a formally real ring; the complex numbers do not, since

$\mathrm{i}^2 + 1^2 = 0 .$

Formally real fields (like that of the real numbers) are particularly interesting; see formally real field.

Created on October 27, 2013 at 16:46:04. See the history of this page for a list of all contributions to it.