How we can prove this
Theorem. Let $R$ be a reduced ring. Then $a\in ZD(R)$ (the zero divisors of $R$) if and only if $a\in P$ for some minimal prime ideal $P$.
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nesreen
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Duplicate of this question and others. See the answers there. There is a search capability that can be used to find questions and answers, see the box in the upper right corner of the page. Also you can use google with site:math.stackexchange.com, which is usually more accurate that the MSE search (but more cumbersome). – Key Ideas May 31 '13 at 19:23
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I want to show if a∈Z(R)then a∈P. Let x∈Z(R). Thus there exists y∈R such that y≠0 and xy=0. Since R is reduced, y∉Nil(R), which is the intersection of all prime ideals of R. hence there exists at least a minimal prime ideal P of R which does not contain y. Now, since xy=0∈P and P is prime, then x∈P. How can I show the inverse? ( if a∈P then a∈Z(R) )
nesreen
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