real ring

A ring $A$ is called real iff the following identity holds for all $n\in \mathbb{N}$ :

$$a_1^2+\mathrm{\dots }+a_n^2=0\iff a_1,\mathrm{\dots },a_n=0\mathit{\hspace{1em}\hspace{1em}}(\forall a_1,\mathrm{\dots },a_n\in A)$$

Remark.

If $A$ is a ring then being real implies the following

• $A$ can have a partial ordering

• $A$ is reduced

Conversely, we note that if $A$ is reduced and can have a partial ordering then $A$ is a real ring.If $A$ is a field then we call it a real field. Similarly we define real domains,real (von Neumann) regular rings, $\mathrm{\dots }$