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 R-algebra generated by two elements x and y, i.e., the noncommutative ring R = R〈x, y〉, formed by the real polynomials in the variables x and y, where x y!=y x, the set I = {f(x, y) element R|f(0, 0) = 0} is a two-sided ideal.

ideal | left ideal | one-sided ideal | right ideal