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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 at 17:31
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    $\begingroup$ Near-duplicate: math.stackexchange.com/questions/1326505/… $\endgroup$ – Eric Wofsey Aug 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 at 18:01
  • $\begingroup$ So mostly it deoends on the convention. But I don't understand the argument behind whole ring not being a prime ideal. In the questions they discussed about 0 being any domain, while not about 0 being an integral domain. So I should consider 0 in general not an integral domain and the proceed further? $\endgroup$ – user631697 Aug 19 at 18:13
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    $\begingroup$ That's not a soft question. $\endgroup$ – Paul Aug 19 at 20:01
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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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