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^{}, $\mathrm{\dots}$
Title  real ring 

Canonical name  RealRing 
Date of creation  20130322 18:51:35 
Last modified on  20130322 18:51:35 
Owner  jocaps (12118) 
Last modified by  jocaps (12118) 
Numerical id  6 
Author  jocaps (12118) 
Entry type  Definition 
Classification  msc 13J30 
Classification  msc 13J25 
Related topic  FormallyRealField 