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Every ring with unity has at least two units

  1. May 2, 2017 #1
    1. The problem statement, all variables and given/known data
    T/F: Every ring with unity has at least two units

    2. Relevant equations


    3. The attempt at a solution
    I thought that the answer was true, because if a ring ##R## has unity ##1##, then ##1 \cdot 1 = 1## and ##(-1) \cdot (-1) = 1##. Where am I going wrong?
     
  2. jcsd
  3. May 2, 2017 #2

    Stephen Tashi

    User Avatar
    Science Advisor

    The elements denoted by "##a##" and "##-a##" are not necessarily distinct from each other. Can you think of a ring where 1 = -1 ?
     
  4. May 3, 2017 #3
    I can only think of the zero ring as an example. Are there non-trivial examples?
     
  5. May 3, 2017 #4

    fresh_42

    Staff: Mentor

    Many. The zero ring hasn't any units at all. So what do you need also to speak of units and what is the smallest example? Based on this, there are plenty of examples.
     
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