A question arising from the Krull intersection theorem. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T22:03:22Z http://mathoverflow.net/feeds/question/43499 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43499/a-question-arising-from-the-krull-intersection-theorem A question arising from the Krull intersection theorem. TmobiusX 2010-10-25T09:11:41Z 2010-10-25T11:37:41Z <p>Let R be a local ring, I an ideal, M a finitely generated module and \$N=\cap _nI^nM\$. Then the Krull intersection theorem states that \$N=IN\$. Now if R is a local ring of characteristic \$p>0\$, for each \$e\geq 0\$ let \$I^e\$ denote the ideal generated by the \$p^e\$-th power of the elements of \$I\$. let \$N=\cap _eI^eM\$. Is it ture that \$N=I^0N\$?</p> http://mathoverflow.net/questions/43499/a-question-arising-from-the-krull-intersection-theorem/43503#43503 Answer by Qing Liu for A question arising from the Krull intersection theorem. Qing Liu 2010-10-25T10:34:46Z 2010-10-25T10:34:46Z <p>First Krull's theorem is for Noetherian (not necessarily local) rings. Let \$n\ge 1\$. If \$I\$ is generated by \$r\$ elements \$x_1, \dots, x_r\$, then the usual \$n\$-power \$I^n\$ of \$I\$ is contained in your \$I^e\$ if \$n/r \ge p^e\$ (any element of \$I^n\$ is a combination of \$x_1^{a_1}...x_r^{a_r}\$ with \$a_1+\dots + a_r=n\$, so \$\max_i{a_r} \ge n/r\$). Therefore your \$N\$ is equal to the usual \$\cap_n I^nM\$ and the answer to your question is yes. </p>