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A ring $R$ is called reduced if it has no nilpotent elements (non-trivial). Assume that $R \neq 0$ and consider the following four conditions on $R$:

  1. the ring $R$ is reduced

  2. the localization $R_{P}$ at any its prime ideal $P \subset R$ is reduced

  3. the ring $R$ is an integral domain

  4. the localization $R_{P}$ at any its prime ideal $P \subset R$ is an integral domain.

Question: for every pair of distinct integers $a , b \in \{1,2,3,4\}$, determine if condition $(a)$ implies $(b)$.

So far, I have been able to show that:

(i) 1 doesn't imply 3 (I gave counter example) and 3 implies 1

(ii) I showed 1 if and only if 2.

(iii) I showed 3 implies 4 but 4 doesn't imply 3.

Our ring R is commutative and contains 1.

Please guys I will be highly happy and grateful if the pairs $(1,4), (4,1), (2,3), (3,2), (2,4, (4,2).$ Counter examples will be great too. Thanks.

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    Yea you are right. Thank you – 2017-02-16
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    Sorry, I think I misnumbered things in my earlier comment. – 2017-02-17
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    You surely know that 4 implies 1, 3 implies 2, and 4 implies 2. You also know that 2 doesn't imply 3. Two more implications left. Hint: both are wrong. – 2017-02-19

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