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  1. If the localization $R_p$ of a ring $R$ at each prime ideal $p$ in A is Noetherian, does this imply that $A$ is Noetherian?

  2. What we call such rings which is not Noetherian but localization at each prime ideal is Noetherian ?

  3. Can somebody provide me any counterexample of (1) and also a good reference?

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    (2) Locally Noetherian. (Google for this.) – 2017-01-22
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    There even exist nonNoetherian rings whose localizations at all primes are all *fields*. (The Boolean ring at the linked question is such a ring.) – 2017-01-22
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    @user26857 Do we have some special name in mathematics literature for such rings? – 2017-01-23
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    @Anoopsingh : maybe "locally noetherian" ? This is just a guess. – 2017-01-30

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Consider the quotient of a polynomial ring in infinitely many variables with coefficients in a field by the ideal generated by all monomials of degree 2