# A question on quotient rings and integral domains.Watch

#1
Hello there,

I know that if R is an integral domain, then it is not necessarily true that R/J is an integral domain. (Where R is a ring and J is an ideal in R with J≠R)

I just considered the example R=the set of integers and J=Z/4Z (Z denoting the set of integers here). So I found that in this case, R/J has zero divisors and so cannot be an integral domain.

However, the converse seems a lot more difficult to prove/disprove. i.e.
"If R/J is an Integral Domain, then so is R". I'm not actually sure if this is true to begin with. If it isn't, I must provide a counterexample.

Regards beast!
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13 years ago
#2
It's false. Take for instance R = Z x Z. Then J = Z x {0} is a (prime) ideal of R and R/J is an integral domain, but R isn't.
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#3
Thanks again!!
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#4
(Original post by dvs)
It's false. Take for instance R = Z x Z. Then J = Z x {0} is a (prime) ideal of R and R/J is an integral domain, but R isn't.
Why isn't R an integral domain. It has a 1≠0, no zero divisors and is commutative. I'm not sure why it isn't??!
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13 years ago
#5
(Original post by beast)
Why isn't R an integral domain. It has a 1≠0, no zero divisors and is commutative. I'm not sure why it isn't??!
Yes it has. e.g. (1,0) x (0,1) = (0,0)
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#6
(Original post by Neapolitan)
Yes it has. e.g. (1,0) x (0,1) = (0,0)
Yes Just a split second before you sent that reply, it registered!! Thanks!!

I've actually provided that as a counterexample before!! ( I think I need to wake up)
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