Editing Order (ring theory) (section)

==Examples==
Some examples of orders are:<ref>Reiner (2003) pp.&nbsp;108–109</ref>
* If <math>A</math> is the [[matrix ring]] <math>M_n(K)</math> over <math>K</math>, then the matrix ring <math>M_n(R)</math> over <math>R</math> is an <math>R</math>-order in <math>A</math>
* If <math>R</math> is an integral domain and <math>L</math> a finite [[separable extension]] of <math>K</math>, then the [[integral closure]] <math>S</math> of <math>R</math> in <math>L</math> is an <math>R</math>-order in <math>L</math>.
* If <math>a</math> in <math>A</math> is an [[integral element]] over <math>R</math>, then the [[polynomial ring]] <math>R[a]</math> is an <math>R</math>-order in the algebra <math>K[a]</math>
* If <math>A</math> is the [[group ring]] <math>K[G]</math> of a [[finite group]] <math>G</math>, then <math>R[G]</math> is an <math>R</math>-order on <math>K[G]</math>

A fundamental property of <math>R</math>-orders is that every element of an <math>R</math>-order is [[integral element|integral]] over <math>R</math>.<ref name=R110>Reiner (2003) p.&nbsp;110</ref>

If the integral closure <math>S</math> of <math>R</math> in <math>A</math> is an <math>R</math>-order then the integrality of every element of every <math>R</math>-order shows that <math>S</math> must be the unique maximal <math>R</math>-order in <math>A</math>.  However <math>S</math> need not always be an <math>R</math>-order: indeed <math>S</math> need not even be a ring, and even if <math>S</math> is a ring (for example, when <math>A</math> is commutative) then <math>S</math> need not be an <math>R</math>-lattice.<ref name=R110/>