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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.2012-12-30
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    @gumti $I:J$ is called the [quotient ideal](http://en.wikipedia.org/wiki/Ideal_quotient) not index.2012-12-30

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