ideal included in union of prime ideals
In the following is a commutative ring with unity.
Let be an ideal of the ring and be prime ideals of . If , for all , then .
We will prove by induction on . For the proof is trivial. Assume now that the result is true for . That implies the existence, for each , of an element such that and . If for some , then we are done. Thus, we may consider only the case , for all .
Let . Since is prime then , for all . Moreover, for , the element . Consider the element . Since and , it follows that , otherwise , contradiction. The existence of the element proves the proposition.∎
Let be an ideal of the ring and be prime ideals of . If , then , for some .
|Title||ideal included in union of prime ideals|
|Date of creation||2013-03-22 16:53:14|
|Last modified on||2013-03-22 16:53:14|
|Last modified by||polarbear (3475)|
|Synonym||prime avoidance lemma|