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If I have to show that ideal A is not maximal, is it enough to show that A is not prime because it is usually easier? Every maximal ideal is prime so if we have ideal that is not prime, it can not be maximal, if I am thinking right.

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    You are correct, though it will not always be possible to show that it is not prime (as there might be non-maximal prime ideals). – Tobias Kildetoft May 02 '16 at 12:12
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    For example consider the polynomial ring $\mathbb{C}[X,Y]$. Let $I$ be the ideal generated by $X$, that is, $I$ contains all elements of the form $g\cdot X$ where $g\in \mathbb{C}[X,Y]$, then $I$ is not maximal but it is a prime ideal. – Mathematician 42 May 02 '16 at 12:30

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