i know that commutative VNR are zero dimensional and reduced.But i cant see the converse inclusion.can anyone help me?
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1if $R$ is reduced then $x^n$ is not zero for any positive integer $n$.so we can write $x^n=xx^{n2}x$, and say $x^{n}$ is regular.but i cant see how i can say $x$ is regular foer every $x$ in $R$.And i dint use that $R$ is zero dimensional. – idem Apr 24 '17 at 09:37

1and on the other way, if $R$ is zero dimensional then every prime maximal and $Nil(R)=Rad(R)$,and since it is reduced $Nil(R)=0=Rad(R)$.So intersection of maximals is zero.Can i write $R$ as a direct sum of these maximal ideals?can i say these sum is finite?if i can,then $R$ is seisimple,and hence VNR. – idem Apr 24 '17 at 09:41

[Also related](https://math.stackexchange.com/a/629307/29335) – rschwieb Apr 24 '17 at 13:25