# real ring

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

 $a_{1}^{2}+\dots+a_{n}^{2}=0\Leftrightarrow a_{1},\dots,a_{n}=0\qquad(\forall a% _{1},\dots,a_{n}\in A)$
###### 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, $\dots$

Title real ring RealRing 2013-03-22 18:51:35 2013-03-22 18:51:35 jocaps (12118) jocaps (12118) 6 jocaps (12118) Definition msc 13J30 msc 13J25 FormallyRealField