Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

im asked to prove that the set consisting of all zero divisors in a commutative ring with unity contains at least one prime ideal.

i cant even start in the proof , ive just defined my set but cant move on construction the ideal !

share|improve this question
    
Nice Question.. Integral domain has $(??)$ as prime ideal... Can you see how and can you think of what would be the case of general commutative ring? –  Praphulla Koushik Jan 4 at 16:59
    
@PraphullaKoushik for integral domains (p) is a prime ideal iff p is a prime number, then? –  Enas Jan 4 at 17:10
    
Check to see that your set is an ideal, then use the fact that $R/I$ is an integral domain if and only if $I$ is prime. By the way I think there are integral domains with prime ideals not generated by a single prime. It holds for principal ideal domains though. –  user111013 Jan 4 at 17:19
add comment

1 Answer

up vote 4 down vote accepted

One way to approach this is to combine these two lemmas:

  1. In a commutative ring, there exist minimal prime ideals.

  2. In a commutative ring with identity, minimal primes consist entirely of zero divisors (I use the convention that $0$ is a zero divisor here.)

share|improve this answer
    
If you don't mind me asking a follow-up question: is it true or false that an ideal which is maximal with respect to being contained in the set of zero divisors is necessarily prime? That's what I started trying to prove... I can post a new question if that's preferred, but I figured I might as well ask you first. –  Dustan Levenstein Jan 4 at 17:33
    
@DustanLevenstein : I keep thinking about this extended question, but I don't have a conclusion either way. A counterexample would have to lie outside of reduced Noetherian rings. In those rings, the zero divisors are a union of prime ideals, and then prime avoidance tells you that an ideal maximal w.r.t. being all zero divisors is exactly one of these ideals. –  rschwieb Jan 5 at 0:21
    
Posted a question. –  Dustan Levenstein Jan 5 at 4:47
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.