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Math Help - Homomorphic image of principal ideal ring

  1. #1
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    Homomorphic image of principal ideal ring

    I'm trying to show that the homomorphic image of a principal ideal ring is again a principal ideal ring. Let \phi:R \to S. If we take a general principal ideal \langle a \rangle and an element of this ideal, ra, then we have \phi(ra) = \phi(r)\phi(a), but this doesn't prove it unless \phi is surjective, correct? Because if it's not surjective then we don't know that every element in S is of the form \phi(r), right? This is what I'm stuck on. Any hints (not answers)?
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  2. #2
    Senior Member Tinyboss's Avatar
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    You want to show that the homomorphic image is a PID. Every function is a surjection onto its image.
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  3. #3
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    φ:R-->φ(R) is always onto.
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  4. #4
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    Yes, true. So do we then have the desired result since subrings are preserved, and \phi(ra)=\phi(r)\phi(a) so that the elements of the image are of the form s\phi(a) with s \in S?
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