A subring <i>A</i> of a ring <i> R</i> is called a (two-sided) ideal of <i>R </i>  if 
   for every <i>r</i> in <i> R</i> and every <i>a </i> in <i> A</i> both <i> ra</i> and <i> ar</i>  are in <i>A</i>.