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Show that two rings are not isomorphic

  1. Mar 22, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that the rings ##2 \mathbb{Z}## and ##3 \mathbb{Z}## are not isomorphic.

    2. Relevant equations


    3. The attempt at a solution
    I know how to show that two structures are isomorphic: find an isomorphism. However, I am not quite sure how to show that there exists no isomorphism at all
     
  2. jcsd
  3. Mar 22, 2017 #2

    phinds

    User Avatar
    Gold Member
    2016 Award

    I don't have any help for you but I have to comment that your post gave me a chuckle because my son and I were just recently discussing a linguistic phenomenon that I noticed some time ago. I make no representation that I know what YOU mean, but I know what I mean, and what most people mean with the following construct:

    "I don't exactly know how to ... " or "I don't quite know how to ... " generally means "I don't have even the tiniest clue how to and in fact I'm not even sure how to spell it" :biggrin:
     
  4. Mar 22, 2017 #3

    fresh_42

    Staff: Mentor

    Assume they were isomorphic and consider the consequences, or try to find a ring homomorphism by defining ##\varphi (2) = 3x## for some integer ##x## and see if you run into contradictions.
     
  5. Mar 23, 2017 #4
    Suppose, for a contradiction, a ring isomorphism [itex]f :2\mathbb{Z}\to3\mathbb{Z}[/itex] existed. Then [itex]f(2) = 3m[/itex] for some [itex]m\in\mathbb{Z}[/itex].
    Since [itex]f[/itex] respects addition and multiplication, then [itex]f(2)+f(2) =f(2+2) =f(4)= f(2\cdot 2)=f(2)\cdot f(2)[/itex]. But this is a problem. Can you explain, why?
     
    Last edited: Mar 23, 2017
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