**Conditions for :**

Let be a multiplicative subset of a commutative ring with identity and let be an ideal of . Then if and only if .

**Proof**

(H pg 146)

Assume . Consider the ring homomorphism given by (for any ). Then hence for some , . Since , we have for some , i.e.\ . But and imply .

If , then . Note that for any , since and . Thus .