Turn on thread page Beta
 You are Here: Home >< Maths

# Is my proof correct? (Rings and Integral domains) watch

1. Hello,
I am wondering whether or not my proof is correct (Please correct me if I am wrong). I must prove the following:

"Suppose that R and S are isomorphic rings. Prove that R is an integral domain if and only if S is an integral domain."

PROOF: (I know that there are two parts to the proof because it's an 'if and only if' proof. This half of the proof aims to show that S is an I.D iff R is an I.D. Hopefully if I can correct this part of the proof the other half will be easy to correct)

Let f,g be elements of S-{0}, where f=Ø(a) for some a in R-{0} and g=Ø(b) for some b in R-{0}.

Since S is an I.D., f.g = g.f ≠ 0 for all f,g in S-{0}.

Hence, Ø(a).Ø(b) ≠ 0 so Ø(ab) ≠ 0

and Ø(b).Ø(a) ≠ 0 so Ø(ba) ≠ 0.

So for all a,b in R-{0}, ab ≠ 0. Hence R has no zero divisors.

We know that S is commutative,
hence Ø(a).Ø(b) = Ø(ab) = Ø(b)Ø(a) = Ø(ba)
and therefore R is commutative.

S has a 1 ≠ 0 .
So, Ø(a).1 = Ø(a).Ø(1) = Ø(a.1).
So R has a 1 ≠ 0.

Hence, S is an I.D iff R is an I.D.
Thank you for your time!
2. Looks OK so far.
3. Doing the "if" implies the "only if" as the inverse of the isomorphism will also be an isomorphism.
4. Ah , I see what you mean. Thanks for the advice!

--------------

Thanks for checking!!

Turn on thread page Beta
TSR Support Team

We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out.

This forum is supported by:
Updated: November 20, 2005
Today on TSR

### Cambridge interviews

Find out which colleges are sending invitations

### University open days

• University of East Anglia
UEA Mini Open Day Undergraduate
Fri, 23 Nov '18
• Norwich University of the Arts
Fri, 23 Nov '18
• Edge Hill University
Sat, 24 Nov '18
Poll
Useful resources

## Make your revision easier

### Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

### How to use LaTex

Writing equations the easy way

### Study habits of A* students

Top tips from students who have already aced their exams

Can you help? Study help unanswered threads

## Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE