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For two ideals $I$ and $J$ in a commutative ring $R$, define $I : J = \{a\in R : aJ \subset I\}$. In the ring $\mathbb{Z}$ of all integers, if $I = 12\mathbb {Z}$ and $J = 8\mathbb {Z}$, find $I : J$.

How should I solve this problem? Can anyone help me please? Thanks for your time.

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Start by solving the relation $a (8 \mathbb{Z}) \subseteq (12 \mathbb{Z})$ for $a$. Find an equivalent condition on $a$ if this one is too strange to solve. P.S. this is usually called the "colon ideal" or sometimes "ideal quotient": index is something different. –  Hurkyl Dec 30 '12 at 3:05
    
@gumti $I:J$ is called the quotient ideal not index. –  user26857 Dec 30 '12 at 10:34
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1 Answer

Hint:

Which integers $a$ have the property that $a\cdot 8$ is a multiple of $12$?

If $a\cdot 8$ is a multiple of $12$, what can you say about $a\cdot x$ where $x$ is any multiple of $8$?

What does that tell you about the ideal $aJ$, where $J=(8)=\{\text{multiples of }8\}$?

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