Prove that when $F$ is a field, $F[x_1 , x_2]$ is not a principal ideal ring. [duplicate]

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  • If F is a field, then $F[x,y]$ is a Principal Ideal Domain?

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This question is from Herstein’s Topic in Algebra, page 166, question No. 8.




Prove that when $F$ is a field, $F[x_1 , x_2]$ is not a principal ideal ring.




I didn't understand, as no answer has been given in this post:
Prove that it is not a Principal Ideal Ring



Any hints/solution will be appreciated.



Thanks in advance




abstract-algebra




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marked as duplicate by Jendrik Stelzner, J.-E. Pin, Lord Shark the Unknown abstract-algebra
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    see this
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    Aug 26 at 10:23














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This question already has an answer here:



  • If F is a field, then $F[x,y]$ is a Principal Ideal Domain?

    3 answers



This question is from Herstein’s Topic in Algebra, page 166, question No. 8.




Prove that when $F$ is a field, $F[x_1 , x_2]$ is not a principal ideal ring.




I didn't understand, as no answer has been given in this post:
Prove that it is not a Principal Ideal Ring



Any hints/solution will be appreciated.



Thanks in advance




abstract-algebra




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marked as duplicate by Jendrik Stelzner, J.-E. Pin, Lord Shark the Unknown abstract-algebra
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  • 1




    see this
    – Chinnapparaj R
    Aug 26 at 10:23












up vote
1
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up vote
1
down vote

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This question already has an answer here:



  • If F is a field, then $F[x,y]$ is a Principal Ideal Domain?

    3 answers



This question is from Herstein’s Topic in Algebra, page 166, question No. 8.




Prove that when $F$ is a field, $F[x_1 , x_2]$ is not a principal ideal ring.




I didn't understand, as no answer has been given in this post:
Prove that it is not a Principal Ideal Ring



Any hints/solution will be appreciated.



Thanks in advance




abstract-algebra




share|cite|improve this question















This question already has an answer here:



  • If F is a field, then $F[x,y]$ is a Principal Ideal Domain?

    3 answers



This question is from Herstein’s Topic in Algebra, page 166, question No. 8.




Prove that when $F$ is a field, $F[x_1 , x_2]$ is not a principal ideal ring.




I didn't understand, as no answer has been given in this post:
Prove that it is not a Principal Ideal Ring



Any hints/solution will be appreciated.



Thanks in advance





This question already has an answer here:



  • If F is a field, then $F[x,y]$ is a Principal Ideal Domain?

    3 answers





abstract-algebra





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share|cite|improve this question




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edited Aug 26 at 10:25









Jendrik Stelzner

7,58221037




7,58221037










asked Aug 26 at 10:15









stupid

670111




670111




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 Aug 26 at 10:51


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.









  • 1




    see this
    – Chinnapparaj R
    Aug 26 at 10:23












  • 1




    see this
    – Chinnapparaj R
    Aug 26 at 10:23







1




1




see this
– Chinnapparaj R
Aug 26 at 10:23




see this
– Chinnapparaj R
Aug 26 at 10:23










1 Answer
1






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Hint:



$F[x_1,x_2]$ is not a principal ideal ring if you can find an ideal that cannot be generated by a single element. Suppose, for the sake of contradiction, that $$(x_1,x_2)=(f)$$ What can you say about $f$?



Strong hint:




$x_1,x_2in (f)Rightarrow f=x_1x_2gRightarrow (x_1,x_2)subseteq (x_1x_2)$. From here, consider what happens with the ideals under the evaluation homomorphism $$phi: F[x_1,x_2]rightarrow F\x_1mapsto 1\x_2mapsto 0$$







share|cite|improve this answer





























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    4
    down vote



    accepted










    Hint:



    $F[x_1,x_2]$ is not a principal ideal ring if you can find an ideal that cannot be generated by a single element. Suppose, for the sake of contradiction, that $$(x_1,x_2)=(f)$$ What can you say about $f$?



    Strong hint:




    $x_1,x_2in (f)Rightarrow f=x_1x_2gRightarrow (x_1,x_2)subseteq (x_1x_2)$. From here, consider what happens with the ideals under the evaluation homomorphism $$phi: F[x_1,x_2]rightarrow F\x_1mapsto 1\x_2mapsto 0$$







    share|cite|improve this answer


























      up vote
      4
      down vote



      accepted










      Hint:



      $F[x_1,x_2]$ is not a principal ideal ring if you can find an ideal that cannot be generated by a single element. Suppose, for the sake of contradiction, that $$(x_1,x_2)=(f)$$ What can you say about $f$?



      Strong hint:




      $x_1,x_2in (f)Rightarrow f=x_1x_2gRightarrow (x_1,x_2)subseteq (x_1x_2)$. From here, consider what happens with the ideals under the evaluation homomorphism $$phi: F[x_1,x_2]rightarrow F\x_1mapsto 1\x_2mapsto 0$$







      share|cite|improve this answer
























        up vote
        4
        down vote



        accepted







        up vote
        4
        down vote



        accepted






        Hint:



        $F[x_1,x_2]$ is not a principal ideal ring if you can find an ideal that cannot be generated by a single element. Suppose, for the sake of contradiction, that $$(x_1,x_2)=(f)$$ What can you say about $f$?



        Strong hint:




        $x_1,x_2in (f)Rightarrow f=x_1x_2gRightarrow (x_1,x_2)subseteq (x_1x_2)$. From here, consider what happens with the ideals under the evaluation homomorphism $$phi: F[x_1,x_2]rightarrow F\x_1mapsto 1\x_2mapsto 0$$







        share|cite|improve this answer














        Hint:



        $F[x_1,x_2]$ is not a principal ideal ring if you can find an ideal that cannot be generated by a single element. Suppose, for the sake of contradiction, that $$(x_1,x_2)=(f)$$ What can you say about $f$?



        Strong hint:




        $x_1,x_2in (f)Rightarrow f=x_1x_2gRightarrow (x_1,x_2)subseteq (x_1x_2)$. From here, consider what happens with the ideals under the evaluation homomorphism $$phi: F[x_1,x_2]rightarrow F\x_1mapsto 1\x_2mapsto 0$$








        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Aug 26 at 10:42

























        answered Aug 26 at 10:20









        cansomeonehelpmeout

        5,2233830




        5,2233830












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