formally real algebra

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

ix i 2=0 \sum_i x_i^2 = 0

(for a finite sum), each x i=0x_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 ABA \to B is a monomorphism of rings (or of rigs, etc), then AA must be formally real if BB is.

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

i 2+1 2=0. \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 16:43:21. See the history of this page for a list of all contributions to it.