The term two-sided ideal is used in noncommutative rings to denote a subset that is both a right ideal and a left ideal. In commutative rings, where right and left are equivalent, a two-sided ideal is simply called "the" ideal.
In the free 
-algebra generated by two elements 
and 
, i.e., the noncommutative
ring 
,
formed by the real polynomials in the variables 
and 
, where 
, the set
 |
(1)
|
is a two-sided ideal. In fact it is an additive subgroup and, for all 
,
 |
(2)
|
i.e.,
 |
(3)
|
This is true even if, in general, 
.
