Fundamental homomorphism theorem (epimorphism)











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0
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Let φ : R → S be a ring epimorphism. Prove that R/kerφ ∼= S.



Is this the fundamental homomorphism theorem? I thought the FHT started with a ring homomorphism and not an epimorphism. Does this change the proof of the theorem ?






ring-theory







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  • If $varphi$ is epi-, then what is $mathrm {Im}, varphi$ in the FHT then?
    – xbh
    Nov 12 at 16:43

















up vote
0
down vote

favorite












Let φ : R → S be a ring epimorphism. Prove that R/kerφ ∼= S.



Is this the fundamental homomorphism theorem? I thought the FHT started with a ring homomorphism and not an epimorphism. Does this change the proof of the theorem ?






ring-theory







share|cite|improve this question






















  • If $varphi$ is epi-, then what is $mathrm {Im}, varphi$ in the FHT then?
    – xbh
    Nov 12 at 16:43















up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let φ : R → S be a ring epimorphism. Prove that R/kerφ ∼= S.



Is this the fundamental homomorphism theorem? I thought the FHT started with a ring homomorphism and not an epimorphism. Does this change the proof of the theorem ?






ring-theory







share|cite|improve this question













Let φ : R → S be a ring epimorphism. Prove that R/kerφ ∼= S.



Is this the fundamental homomorphism theorem? I thought the FHT started with a ring homomorphism and not an epimorphism. Does this change the proof of the theorem ?






ring-theory




ring-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 12 at 16:24









Johnmallu

266




266












  • If $varphi$ is epi-, then what is $mathrm {Im}, varphi$ in the FHT then?
    – xbh
    Nov 12 at 16:43




















  • If $varphi$ is epi-, then what is $mathrm {Im}, varphi$ in the FHT then?
    – xbh
    Nov 12 at 16:43


















If $varphi$ is epi-, then what is $mathrm {Im}, varphi$ in the FHT then?
– xbh
Nov 12 at 16:43






If $varphi$ is epi-, then what is $mathrm {Im}, varphi$ in the FHT then?
– xbh
Nov 12 at 16:43












2 Answers
2






active

oldest

votes

















up vote
0
down vote



accepted










The general formula for the homomorphism theorem is
$$R/ker(phi)cong mathrm{Im}(phi)$$
In the special case, $phi$ is an epimorphism, then $mathrm{Im}(phi)=S$, but be careful, you cannot deduce the general formula from the special case, for example when $phi$ isn‘t an epimorphism






share|cite|improve this answer






























    up vote
    0
    down vote













    You can state the theorem without the epimorphism assumption replacing $S$ by $mathrm{Im} varphi$. So both formulation are equivalent.






    share|cite|improve this answer





















    • Okay thank you !
      – Johnmallu
      Nov 12 at 16:52











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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    0
    down vote



    accepted










    The general formula for the homomorphism theorem is
    $$R/ker(phi)cong mathrm{Im}(phi)$$
    In the special case, $phi$ is an epimorphism, then $mathrm{Im}(phi)=S$, but be careful, you cannot deduce the general formula from the special case, for example when $phi$ isn‘t an epimorphism






    share|cite|improve this answer



























      up vote
      0
      down vote



      accepted










      The general formula for the homomorphism theorem is
      $$R/ker(phi)cong mathrm{Im}(phi)$$
      In the special case, $phi$ is an epimorphism, then $mathrm{Im}(phi)=S$, but be careful, you cannot deduce the general formula from the special case, for example when $phi$ isn‘t an epimorphism






      share|cite|improve this answer

























        up vote
        0
        down vote



        accepted







        up vote
        0
        down vote



        accepted






        The general formula for the homomorphism theorem is
        $$R/ker(phi)cong mathrm{Im}(phi)$$
        In the special case, $phi$ is an epimorphism, then $mathrm{Im}(phi)=S$, but be careful, you cannot deduce the general formula from the special case, for example when $phi$ isn‘t an epimorphism






        share|cite|improve this answer














        The general formula for the homomorphism theorem is
        $$R/ker(phi)cong mathrm{Im}(phi)$$
        In the special case, $phi$ is an epimorphism, then $mathrm{Im}(phi)=S$, but be careful, you cannot deduce the general formula from the special case, for example when $phi$ isn‘t an epimorphism







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 18 at 13:27

























        answered Nov 12 at 18:45









        Fakemistake

        1,635815




        1,635815






















            up vote
            0
            down vote













            You can state the theorem without the epimorphism assumption replacing $S$ by $mathrm{Im} varphi$. So both formulation are equivalent.






            share|cite|improve this answer





















            • Okay thank you !
              – Johnmallu
              Nov 12 at 16:52















            up vote
            0
            down vote













            You can state the theorem without the epimorphism assumption replacing $S$ by $mathrm{Im} varphi$. So both formulation are equivalent.






            share|cite|improve this answer





















            • Okay thank you !
              – Johnmallu
              Nov 12 at 16:52













            up vote
            0
            down vote










            up vote
            0
            down vote









            You can state the theorem without the epimorphism assumption replacing $S$ by $mathrm{Im} varphi$. So both formulation are equivalent.






            share|cite|improve this answer












            You can state the theorem without the epimorphism assumption replacing $S$ by $mathrm{Im} varphi$. So both formulation are equivalent.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Nov 12 at 16:47









            Can I play with Mathness

            3894




            3894












            • Okay thank you !
              – Johnmallu
              Nov 12 at 16:52


















            • Okay thank you !
              – Johnmallu
              Nov 12 at 16:52
















            Okay thank you !
            – Johnmallu
            Nov 12 at 16:52




            Okay thank you !
            – Johnmallu
            Nov 12 at 16:52


















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