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We know that if $R $ is a domain then any localization of $R $ at any multiplicative subset of $R $ is indecomposable, that is, has no non trivial idempotents. Now let $R $ be a commutative ring with 1 that has no non trivial idempotents. I am looking for an additional condition on $R $ under which any localization $R_r $ to be indecomposable for each non nilpotent element $r\in R $.

Thanks for any help.

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    $\begingroup$ This is true if and only if $R$ has a unique minimal prime ideal. $\endgroup$
    – Mohan
    Commented May 25, 2020 at 19:49
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    $\begingroup$ @Mohan, thank you, can you expain a little more. Especialy for the only part. $\endgroup$
    – My. A
    Commented May 26, 2020 at 5:27
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    $\begingroup$ @Mohan, I can prove that if $R $ has finite many minimal prime ideals, then the property holds. But I cannt prove the converse. Are you sure that the converse is true? Can you give some hints? $\endgroup$
    – My. A
    Commented May 27, 2020 at 16:37
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    $\begingroup$ I apologize. I am `Noetherian' person and there are only finitely many minimal primes. I should have stressed that. $\endgroup$
    – Mohan
    Commented May 27, 2020 at 17:39

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