Remark
Suppose that R is a non-commutative ring. Then there are three notions of ideal:
Left ideal
Right ideal
Two-sided ideal
A nonempty subset I of
R such that, for all a, b 
I and all r
R,
a + b I,
ra I.
I and all r
R,
I,
ra I.
A nonempty subset I of
R such that, for all a, b I and all r
R,
a + b I,
ar I.
I and all r
R,
I,
ar I.
A subset of R that is both left and right ideal.