nLab formally real algebra

Idea

When working with algebras (possibly associative, possibly nonassociative) over the real numbers, we often want to express the fact that $-1$ has no square root in the algebra. However, this by itself is not enough, since the algebra may lack square roots for other reasons, so we say that $-1$ is not a sum of squares. Even this may be too weak if the algebra has too little division, because we also want to ensure that other negative squares are not sums of squares. So the really general statement is that $0$ is not a sum of squares, except as a sum of squares of $0$ itself.

At this point, we notice that the $\mathbb{R}$-linear structure is irrelevant, and the concept applies to more general rings (possibly nonassociative). Even the existence of opposites is not needed, and the concept applies to (possibly nonassociative) rigs. Really, we just need multiplication, addition, and zero.

Definitions

Let $A$ be ring, or more generally a magma in the category of abelian monoids. Then $A$ is formally real if, whenever

$$\sum_i x_i^2 = 0$$

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

Given an involution, we have a generalization: a $*$-ring is formally complex if, whenever

$$\sum_i x_i^* x_i = 0$$

(for a finite sum), each $x_i = 0$. (Then a ring is formally real iff it is formally complex when equipped with the trivial involution.)

Examples

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

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

However, the complex numbers are formally complex. (These examples are the source of the names.)

Actually, all of the Cayley–Dickson algebras are formally complex. (This is because $x^* x$ is always a real number and is $0$ only when $x$ is.)

The trivial ring is formally real. Every other formally real (or formally complex) ring is infinite (because $1 = 1^2$, $2 = 1^2 + 1^2$, $3 = 1^2 + 1^2 + 1^2$, etc are all distinct). Even without an identity (and even in the nonassociative case), if the ring is formally real and nontrivial, then it is infinite. (If $x \ne 0$, consider $x^2$, $x^2 + x^2$, $x^2 + x^2 + x^2$, etc.)

There are finite formally real rigs, however, such as the boolean rig $\{0, 1\}$. Indeed, any distributive lattice (with a bottom element), viewed as a rig, is formally real.

Among Jordan algebras, the formally real ones are especially important; it is these that (over the real numbers, in finite dimensions) have a nice classification theorem?.

Properties

The trivial algebra is the only formally real algebra over any ring that is not formally real; more generally, if $A \to B$ is a monomorphism of rings (or of rigs, etc), then $A$ must be formally real if $B$ is. Similarly, if $A \to B$ is a monomorphism of $*$-rings, then $A$ must be formally complex if $B$ is.

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

Every formally real (or formally complex) ring has a natural partial ordering: $a \leq b$ iff $b - a$ can be written as a sum of squares (or a sum of terms of the form $x^* x$).