Editing Ordered ring (section)

== Basic properties ==
For all ''a'', ''b'' and ''c'' in ''R'':
*If ''a'' ≤  ''b'' and 0 ≤  ''c'', then ''ac'' ≤  ''bc''.<ref>OrdRing_ZF_1_L9</ref> This property is sometimes used to define ordered rings instead of the second property in the definition above.
*|''ab''| = |''a''|{{Hair space}}|''b''|.<ref>OrdRing_ZF_2_L5</ref>
*An ordered ring that is not [[trivial ring|trivial]] is infinite.<ref>ord_ring_infinite</ref>
*Exactly one of the following is true: ''a'' is positive, −''a'' is positive, or ''a'' = 0.<ref>OrdRing_ZF_3_L2, see also OrdGroup_decomp</ref> This property follows from the fact that ordered rings are [[abelian group|abelian]], [[linearly ordered group]]s with respect to addition. 
*In an ordered ring, no negative element is a square:<ref>OrdRing_ZF_1_L12</ref> Firstly, 0 is square. Now if ''a'' ≠ 0 and ''a'' = ''b''<sup>2</sup> then ''b'' ≠ 0  and ''a''  = (−''b'')<sup>2</sup>; as either ''b'' or −''b'' is positive, ''a'' must be nonnegative.