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Ideal and Factor Ring

  1. Apr 16, 2009 #1
    1. The problem statement, all variables and given/known data

    a) If A is an ideal of R and B is an ideal of S.
    Show that A x B is an ideal of R x S.

    b) Show that every ideal C of RxS has the form C = AxB as in(a)
    [hint: A = { a in R | (a,0) in C}]



    3. The attempt at a solution
    a)Since A and B are ideal of R and S, aR and Ra are subsets of A, bS and Sb are subsets of B.
    Let (a,b) in AxB and (r,s) in RxS, (a,b)(r,s) = (ar,bs) in AxB since ar in A and bs in B.

    b) Let A = { a in R | (a,0) in C} and
    B = { b in S | (0,b) in C}
    We need to show that AxB = <(a,0),(0,b)>.
    my idea is correct?
     
  2. jcsd
  3. Apr 17, 2009 #2
    (a) looks good if you already have a theorem that A x B is a ring. If you don't have that theorem, then don't forget to do that routine step.

    For (b), you have to show [tex]A\times B\subset C[/tex] and [tex]C\subset A\times B[/tex]

    I regard [tex]A\times B=\langle(a,0),(0,b)\rangle\subset C[/tex] as the easier direction because you only need to use the fact that C is a ring. Really there's nothing to show.

    To prove [tex]C\subset A\times B[/tex], I think you need to use the facts that C is an ideal and furthermore that R and S have identities. (Are you allowed to assume R and S have identities? I hope so.)
     
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