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Math Help - ring

  1. #1
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    ring

    Show that an ideal is proper if and only if it does not contain 1.
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  2. #2
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    Prove these two statements: If 1 is in the ideal, it is not proper. If 1 is not in the ideal, it is proper. The second statement is obvious by the definition of being a proper ideal, i.e. if the ring is R and ideal is I, I\neq R since 1\in R, 1 \notin I.

    Use the definition of an (left) ideal: for all elements r of the ring, rI\subseteq I. In particular, if i\in I then ri\in I for any r. What if i=1?
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  3. #3
    Super Member Gamma's Avatar
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    Does your ring definition require all rings to be unitary?

    If you take the ring to be 2\mathbb{Z}. Certainly 2\mathbb{Z}\subset 2\mathbb{Z} and it does not contain 1, but is not a proper ideal since it is the whole ring.
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