A question on quotient rings and integral domains. Watch

beast
Badges: 0
Rep:
?
#1
Report Thread starter 13 years ago
#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.

Please help! Any advice/help would be very much appreciated!

Regards beast!
0
quote
reply
dvs
Badges: 10
Rep:
?
#2
Report 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.
0
quote
reply
beast
Badges: 0
Rep:
?
#3
Report Thread starter 13 years ago
#3
Thanks again!!
0
quote
reply
beast
Badges: 0
Rep:
?
#4
Report Thread starter 13 years ago
#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??! :confused:
0
quote
reply
Neapolitan
Badges: 1
Rep:
?
#5
Report 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??! :confused:
Yes it has. e.g. (1,0) x (0,1) = (0,0)
0
quote
reply
beast
Badges: 0
Rep:
?
#6
Report Thread starter 13 years ago
#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)
0
quote
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise

University open days

  • University of Buckingham
    Psychology Taster Tutorial Undergraduate
    Fri, 14 Dec '18
  • University of Lincoln
    Mini Open Day at the Brayford Campus Undergraduate
    Wed, 19 Dec '18
  • University of East Anglia
    UEA Mini Open Day Undergraduate
    Fri, 4 Jan '19

Were you ever put in isolation at school?

Yes (142)
27.41%
No (376)
72.59%

Watched Threads

View All