with an arbitrary , and then identify with 0 all elements of the ideal It turns out that the ideal xR is the smallest ideal that contains x, called the ideal generated by x. More generally, we can start with an arbitrary subset , and then identify with 0 all the elements in the ideal generated by S: the smallest ideal (S) such that . The ring that we obtain after the identification depends only on the ideal (S) and not on the set S that we started with. That is, if , then the resulting rings will be the same. Therefore, an ideal I of a commutative ring R captures canonically the information needed to obtain the ring of elements of R modulo a given subset . The elements of I, by definition, are those that are congruent to zero, that is, identified with zero in the resulting ring. The resulting ring is called the quotient of R by I and is denoted R/I. Intuitively, the definition of an ideal postulates two natural conditions necessary for I to contain all elements designated as "zeros" by R/I: I is an additive subgroup of R: the zero 0 of R is a "zero" , and if and are "zeros", then is a "zero" too. Any multiplied by a "zero" is a "zero" . It turns out that the above conditions are also sufficient for I to contain all the necessary "zeros": no other elements have to be designated as "zero" in order to form R/I. (In fact, no other elements should be designated as "zero" if we want to make the fewest identifications.) Remark. The above construction still works using two-sided ideals even if R is not necessarily commutative. Examples and properties (For the sake of brevity, some results are stated only for left ideals but are usually also true for right ideals with appropriate notation changes.) In a ring R, the set R itself forms a two-sided ideal of R called the unit ideal. It is often also denoted by since it is precisely the two-sided ideal generated (see below) by the unity . Also, the set consisting of only the additive identity 0R forms a two-sided ideal called the zero ideal and is denoted by . Every (left, right or two-sided) ideal contains the zero ideal and is contained in the unit ideal. An (left, right or two-sided) ideal