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Thread: Ideal in a conmmutative ring

  1. #1
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    Ideal in a conmmutative ring

    Hi! I have problems with this demostration


    Let $\displaystyle I$ be an ideal in a commutative ring $\displaystyle R$. If $\displaystyle J$ is an ideal in $\displaystyle R$ and $\displaystyle I\subseteq{J}$, prove that:


    $\displaystyle J/I = \{ r+I : r \in{J} \} $ is an ideal in $\displaystyle R/I$


    Demostration

    If $\displaystyle J+I\in J/I$, then $\displaystyle J+I\in R/I$ because $\displaystyle J\subset R$, therefore $\displaystyle J/I\subset R/I$.

    Is the demonstration correct?
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  2. #2
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    Re: Ideal in a conmmutative ring

    Quote Originally Posted by cristianoceli View Post
    Hi! I have problems with this demostration
    Let $\displaystyle I$ be an ideal in a commutative ring $\displaystyle R$. If $\displaystyle J$ is an ideal in $\displaystyle R$ and $\displaystyle I\subseteq{J}$, prove that: $\displaystyle J/I = \{ r+I : r \in{J} \} $ is an ideal in $\displaystyle R/I$
    Demostration
    If $\displaystyle J+I\in J/I$, then $\displaystyle J+I\in R/I$ because $\displaystyle J\subset R$, therefore $\displaystyle J/I\subset R/I$.
    Is the demonstration correct?
    By Demostration I must assume that you mean PROOF.
    You ask: Is the demonstration correct?
    That depends upon what is required. What you have posted is correct under certain conditions.
    We have no way of knowing what your lecturer requires as a proof.
    For me, I expect a student to demonstrate that all conditions for $J/I$ to be an ideal in $R/I$ are meet.
    Have you done that?
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