What are the most simple examples of a commutative ring $R$ satisfying both of the following two properties:
1. $R$ is not Noetherian.
2. $R$ has exactly one prime ideal.
Non-Noetherian ring with a single prime ideal
10
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abstract-algebra
commutative-algebra
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3Hint: you can get an example of this by finding a ring whose maximal ideal consists of nilpotent elements but is not nilpotent (e.g. a suitable quotient of $k[x_1, x_2, \dots]$). – 2011-05-16
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0@Akhil: I was trying to look at a quotient of the form k[x1,...]/(x1, x2^2,...xn^n,...), but it doesn't seem to work. An additional hint would be highly appreciated. – 2011-05-16
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1@spec: Dear Spec, Actually, what you just described is precisely the example I had in mind; that ring has one prime ideal (namely, that generated by all the $x_i$), which is not nilpotent. – 2011-05-16