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I need to think of an example of a ring R with ideals I and J where I is an ideal of J and J is an ideal of R, but I is not an ideal of R.

Kevin
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    https://math.stackexchange.com/questions/2068158/a-finite-associative-ring-with-non-transitive-ideals – Dave Jun 07 '17 at 00:26
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    https://math.stackexchange.com/questions/144022/subideals-of-an-ideal – Dave Jun 07 '17 at 00:26

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