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Ring with Infinitely Many Simple Modules

  1. Apr 10, 2013 #1
    1. The problem statement, all variables and given/known data
    Give an example of a ring [itex] R[/itex] with infinitely many non-isomorphic simple modules.
    3. The attempt at a solution
    I was thinking of setting
    [itex]R=\mathbb{Z}_{p_1}\times \mathbb{Z}_{p_2}\times \mathbb{Z}_{p_3}\times \cdots [/itex]
    where [itex] p_1,p_2,p_3,\ldots [/itex] is an infinite increasing list of distinct prime numbers. Then each [itex]\mathbb{Z}_{p_i}[/itex] is an ideal of the ring and each [itex]\mathbb{Z}_{p_i}[/itex] is in fact simple because it is generated by any non-zero element.

    Is this correct? Can anybody think of another (possibly better) example?
  2. jcsd
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