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Thread: 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 $\displaystyle R$ and ideal is $\displaystyle I$, $\displaystyle I\neq R$ since $\displaystyle 1\in R, 1 \notin I$.

    Use the definition of an (left) ideal: for all elements $\displaystyle r$ of the ring, $\displaystyle rI\subseteq I$. In particular, if $\displaystyle i\in I$ then $\displaystyle ri\in I$ for any $\displaystyle r$. What if $\displaystyle i=1$?
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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 $\displaystyle 2\mathbb{Z}$. Certainly $\displaystyle 2\mathbb{Z}\subset 2\mathbb{Z}$ and it does not contain 1, but is not a proper ideal since it is the whole ring.
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