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For a subgroup $(I,+)$ of additive group part of ring $(R,+,•)$ to be an Ideal we need $a•I$ and $I•a$ to be subset of $I$ for all $a$ in $R$.

The exercises in my textbook gives a lot of examples of ideals, but I want to see examples of those subgroups of $(R,+)$ which are not Ideals.

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1 Answer

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$\mathbb{R}$ is a field, hence has only the zero and unit ideals, however it has lots of non-trivial subgroups. For instance, $\mathbb{Z}$ and $\mathbb{Q}$.

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