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Here is a soft question that I am dealing with. Please tell me if it's correct or not.

Suppose $A$ is a commutative ring with unity. Is $A$ a prime ideal of $A$?

I think the answer is true, because we know $I$ is an integral domain iff $R/I$ is an integral domain. But $A/A = 0.$ So it's an integral domain. So$A$ is a prime ideal in $A$ Tell me if I argument is right or wrong. Any help will be appreciated. Thanks

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    $\begingroup$ Usually when we consider prime ideals, we define them to be proper ideals, i.e. ideals that are not the whole of $A$. $\endgroup$ – rubikscube09 Aug 19 '19 at 17:31
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    $\begingroup$ Near-duplicate: math.stackexchange.com/questions/1326505/… $\endgroup$ – Eric Wofsey Aug 19 '19 at 17:58
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    $\begingroup$ Also relevant: math.stackexchange.com/questions/427078/is-0-a-field $\endgroup$ – Eric Wofsey Aug 19 '19 at 18:01
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    $\begingroup$ That's not a soft question. $\endgroup$ – Paul Aug 19 '19 at 20:01
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    $\begingroup$ @user631697 you could call it a convention, but if so, this is really one of those conventions that has nearly 100% of the usage. I have never seen the alternative used even once in a text or paper. $\endgroup$ – rschwieb Aug 20 '19 at 2:35
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It's usually taken as part of the definition that an integral domain must not be the zero ring.

For example, on Wikipedia (first sentence of the article, note the word “nonzero”), and in the standard references (Atiyah & Macdonald, Matsumura, Lang, etc.).

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