ideal of R, then is a left ideal of the subring of S: unless f is surjective, need not be an ideal of S; see also #Extension and contraction of an ideal below. Ideal correspondence: Given a surjective ring homomorphism , there is a bijective order-preserving correspondence between the left (resp. right, two-sided) ideals of containing the kernel of and the left (resp. right, two-sided) ideals of : the correspondence is given by and the pre-image . Moreover, for commutative rings, this bijective correspondence restricts to prime ideals, maximal ideals, and radical ideals (see the Types of ideals section for the definitions of these ideals). (For those who know modules) If M is a left R-module and a subset, then the annihilator of S is a left ideal. Given ideals of a commutative ring R, the R-annihilator of is an ideal of R called the ideal quotient of by and is denoted by ; it is an instance of idealizer in commutative algebra. Let be an ascending chain of left ideals in a ring R; i.e., is a totally ordered set and for each . Then the union is a left ideal of R. (Note: this fact remains true even if R is without the unity 1.) The above fact together with Zorn's lemma proves the following: if is a possibly empty subset and is a left ideal that is disjoint from E, then there is an ideal that is maximal among the ideals containing and disjoint from E. (Again this is still valid if the ring R lacks the unity 1.) When , taking and , in particular, there exists a left ideal that is maximal among proper left ideals (often simply called a maximal left ideal); see Krull's theorem for more. An arbitrary union of ideals need not be an ideal, but the following is still true: given a possibly empty subset X of R, there is the smallest left ideal containing X, called the left ideal generated by X and is denoted by . Such an ideal exists since it is the intersection of all left ideals containing X. Equivalently, is the set of all the (finite) left R-linear combinations of elements of X over R: (since such a span is the smallest left ideal containing X.) A right (resp. two-sided) ideal generated by X is defined in the similar way. For "two-sided", one has to use