Preimage of a homomorphism

1. Feb 16, 2010

tom.young84

"Preimage" of a homomorphism

f:R-->S is a homomorphism of rings and suppose J is an ideal of S.

Prove that f^-1[J]={r in R: f(r) in J} is an ideal of R.

I'm more concerned about how to even start this proof as I am lost.

2. Feb 16, 2010

Hurkyl

Staff Emeritus
Re: "Preimage" of a homomorphism

Then do something -- even if you don't know it will help you prove the theorem.

Definitions are pretty much always someplace to start. Trying to turn a problem into algebra is pretty much always something to try. Specific examples are often useful.

You should never have no idea how to start working on a problem.

3. Feb 16, 2010

Dick

Re: "Preimage" of a homomorphism

You want to show that if x is any element of R and j is any element of f^(-1)(J) then xj is also an element of f^(-1)(J). Is that enough of a start?

4. Feb 16, 2010

tom.young84

Re: "Preimage" of a homomorphism

I get how to show something is in f-1[J] through absorption.

Suppose an r in R and j in J

f(rj)=f(r)f(j) which is in J

I'm stuck on show that f-1[J] is an ideal by showing that addition is closed and that there are inverses. I know 0 is in it since homomorphism carries 0R to 0S and 0S is in the ideal.

5. Feb 16, 2010

Dick

Re: "Preimage" of a homomorphism

I thought you said you didn't know how to start. Why don't you try to use the properties of homomorphism to show f^(-1)(J) is closed under addition. After all, J is.

Last edited: Feb 16, 2010
6. Feb 16, 2010

tom.young84

Re: "Preimage" of a homomorphism

Is this allowed?

r1, r2 in R

f(r1+r2)=f(r1)+f(r2) in f^-1[J] or would I have to toss in a f(j) such that
f(r1+r2)f(j)=f(r1)f(j)+f(r2)f(j) in f^-1[J] (this makes use of both absorption and the property of homomorphisms)

7. Feb 16, 2010

Dick

Re: "Preimage" of a homomorphism

No, no, no. You want to start with r1 and r2 in f^(-1)(J). Not r1 and r2 in R. Then prove r1+r2 is in f^(-1)(J). You want to prove f^(-1)(J) is an ideal.

8. Feb 16, 2010

tom.young84

Re: "Preimage" of a homomorphism

I keep wanting to pull the idea of homomorphisms into this

so we have this r1, r2 in f-1[J]

f(r1+r2)=f(r1)+f(r2) in f-1[J]

This shows r1+r2 is in f-1[J]

Thus r1+r2 is in f-1[J]

Last edited: Feb 16, 2010
9. Feb 16, 2010

Dick

Re: "Preimage" of a homomorphism

You should probably get a piece of blank paper out and write down all of the definitions of everything here so you can look at them all at once. f(r1) isn't in f^(-1)(J). It's in J.

10. Feb 17, 2010

tom.young84

Re: "Preimage" of a homomorphism

These is the chunk of the proof i was having difficulty with, I think I corrected it.

r1, r2 in R and j in J

f(r1+r2)=f(r1)+f(r2) in J (definition of homomorphism)
hence, r1+r2 f-1(J).

Inverses
-r in R

f(-r)=-f(r) in J
hence -r in f-1(J)

Absorption
f(r1r2)f(j)=f(r1)f(r2)f(j) in J (Using the definition of homomorphism)
hence r1r2j=(r1r2)j=r1(r2j) in f-1(J) by absorption

Yay ideal

11. Feb 17, 2010

Dick

Re: "Preimage" of a homomorphism

I THINK you know what you're doing. Again, I would make your starting point, "assume r1 and r2 are in f^(-1)(J)", not "assume r1 and r2 are in R". I already said this but you want to prove f^(-1)(J) is closed, not that R is closed. You already know that. Sure, and then use properties of homomorphism and that J is an ideal of S.

12. Feb 17, 2010

tom.young84

Re: "Preimage" of a homomorphism

So my proofs should go something like

r1, r2 in f-1(J) => Definition of homomorphism in J => operation is closed in f-1(J)

This problem is giving me a hard time. Last semester I butched a similar problem that dealt with the preimage of a subgroup.

13. Feb 17, 2010

Dick

Re: "Preimage" of a homomorphism

It's like at least 80% right. What I'm missing a little precision. Like EXACTLY what set something is coming from and what the properties of that set are that make your statement true. Like look at your "Inverses" part. You say (-r) in R, then you say f(-r)=(-f(r)), ok so far but then you say -f(r) in J. That's wrong. That ONLY TRUE if you start with r in f^(-1)(J) not just in R. Do you see what I'm saying??

14. Feb 17, 2010

tom.young84

Re: "Preimage" of a homomorphism

I'm starting to see what you are saying. I'm reworking having addition closed.

r1, r2 in f^(-1)(J)

f(r1+r2)=f(r1)+f(r2) which is in J and hence r1+r2 is in f^(-1)(J)

15. Feb 17, 2010

Dick

Re: "Preimage" of a homomorphism

You've got the idea. And that's true because f(r1) and f(r2) are in J. And J is closed, being an ideal and all.

16. Feb 18, 2010

tom.young84

Re: "Preimage" of a homomorphism

I just want to formally prove this so as to make a good answer hopefully available.

Nonempty:
f(0) is in J (since its an ideal)
Hence 0 is in f-1(J)

Suppose an r1, r2 in f-1(J).
f(r1+r2)=f(r1)+f(r2) which is in J.
Hence r1+r2 is in f-1(J)

Inverses:
We know that for each f(r) in J, there exists a -f(r) in J (since J is an ideal).
By a few lemmas we have on negatives and homomorphisms -f(r)=f(-r).
Hence -r is in f-1(J).

Absorption:
i is in f-1(J)
f(ri)=f(r)f(i); we know that f(i) is in J; by using the definition of homomorphism f(ri) is in J, so ri is in f-1(J)

Likewise for the other direction since we are assuming R and S are general rings.

17. Feb 19, 2010

Dick

Re: "Preimage" of a homomorphism

You are still loosing a bit focus on what you are trying to prove. Look at absorption. You want to pick i in f^(-1)(J) and r in R and prove i*r is in f^(-1)(J), right? Do you see what's wrong with your proof??