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Rings with unity

  1. Feb 8, 2008 #1
    [SOLVED] rings with unity

    1. The problem statement, all variables and given/known data
    Corollary 27.18 (in Farleigh) tells us that every ring with unity contains a subring isomorphic to either Z or some Z_n. Is it possible that a ring with unity may simultaneously contain two subrings isomorphic to Z_n and Z_n with n not equal to m? If it is possible, give an example. If it is impossible, prove it.

    EDIT: change the second Z_n to Z_m
    2. Relevant equations



    3. The attempt at a solution
    My intuition tells me it is impossible. But I have no idea how to prove it.
     
    Last edited: Feb 9, 2008
  2. jcsd
  3. Feb 9, 2008 #2
    What about Z/8Z? All of its subrings are isomorphic to some Zn. Does that mean all its subrings are isomorphic to each other?
     
  4. Feb 9, 2008 #3

    Hurkyl

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    ehrenfest: Given that the meanings you give to the words is not the most common, you really need to be specific about their meaning.

    e.g. I have absolutely no idea if, in this particular context, your use of 'subring' in this context requires the subring to contain a unit, and if that unit has to be the same as the one in the enclosing ring.
     
  5. Feb 9, 2008 #4
    A ring <R,+,*> is a set R together with two binary operations + and * such that the following axioms are satisfied:
    1) <R,+> is an abelian group
    2) Multiplication is associative
    3) For all a,b,c in R, a*(b+c)=(b+c)*a=a*b+a*c

    A subring is a subset of a ring that is also a ring.

    See the EDIT.

    Yes. I see. Z_8, we have the subring {0,4} which isomrphic to Z_2 and the whole ring is also a subring.
     
    Last edited: Feb 9, 2008
  6. Feb 9, 2008 #5

    Hurkyl

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    The subalgebra {0, 4} of Z_8 is not isomorphic to Z_2.
     
    Last edited: Feb 9, 2008
  7. Feb 9, 2008 #6
    Why?
    4+4=0
    0+4=4
    0+0=0
    4*4=0
    4*0=0
    0*0=0

    This is the same algebra as Z_2.

    EDIT: you're right 1*1=1 not 0
    EDIT: then what subring of Z_8 is isomorphic to Z_n where n is not equal to 8?
     
    Last edited: Feb 9, 2008
  8. Feb 9, 2008 #7
    Oops, that example doesn't work. Try Z_4 X Z_2.
     
  9. Feb 9, 2008 #8
    Then we have subrings isomorphic to Z_2 and Z_4. Thanks.
     
  10. Feb 9, 2008 #9
    is the ring Zn the same as the ring Z/Zn? (the one whose elements are modulo equivalence classes, with regular summation and multiplication?)
     
  11. Feb 9, 2008 #10
    Yes, they are isomorphic since the kernel of

    phi: Z -> Z_n defined by phi(z) = z mod n

    is Zn.
     
  12. Feb 9, 2008 #11
    no I meant if Zn was defined in that way >_>

    but it's not ... but almost. Thanks anyway. =)
     
  13. Feb 9, 2008 #12
    What do you mean?
     
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