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 Jun 5 comment Why is it Artinian? I didn't know the alternative definition. Thank you for that. Jun 5 comment Why is it Artinian? Thank you for your help. I understand that for each product $x_1 x_2 \cdots x_n$ where $x_i \in m_i$ there exists $i$ such that $x_i \in p$. But how do we know that $p \supset m_i$ for some $i$. Jun 5 revised Why is it Artinian? added 2 characters in body Jun 5 asked Why is it Artinian? May 6 comment Proof of a corollary of the Noether normalisation lemma @Cantlog : There is an addendum between the lemma and the corollary saying "under the condition of the lemma, if furthermore $k$ is algebraiccaly closed, and $A$ is an integral domain with field of fractions $K$ then ...". So, they may be the additional conditions. May 5 asked Proof of a corollary of the Noether normalisation lemma Apr 11 accepted If $X \times \{0\} \cup A \times I$ is closed in $X \times I$. Then, is $A$ closed in $X$? Apr 11 comment If $X \times \{0\} \cup A \times I$ is closed in $X \times I$. Then, is $A$ closed in $X$? Am I right in thinking that $B\times (0,1]$ is the complement of $X \times \{0\} \cup A \times I$ in $X \times I$. If it is true, then it follows that $B\times (0,1]$ is open. Apr 11 comment Is the length of the composition series of a free module identical to the number of its bases? It is the page 118 of 1969 version. Apr 11 comment Is the length of the composition series of a free module identical to the number of its bases? It works. Then, the description of the example in this famous book is not correct. Anyway, thank you for your help. Apr 11 asked If $X \times \{0\} \cup A \times I$ is closed in $X \times I$. Then, is $A$ closed in $X$? Apr 11 comment Is the length of the composition series of a free module identical to the number of its bases? In the example, $A_n$ is the set of the homogeneous polynomials of degree $n$. Apr 11 comment Is the length of the composition series of a free module identical to the number of its bases? $A$ is a Noetherian graded ring $A=\oplus_{n=0}^\infty A_n$. $M$ is a finitely-generated graded $A$-module $M=\oplus_{n=0}^\infty M_n$. Apr 11 revised Is the length of the composition series of a free module identical to the number of its bases? added 4 characters in body Apr 11 revised Is the length of the composition series of a free module identical to the number of its bases? added 4 characters in body Apr 11 asked Is the length of the composition series of a free module identical to the number of its bases? Mar 25 comment If a chain of distinct irreducible closed subsets of a quasi-affine variety $Y$ is maximal, a chain of their closures is maximal in $\overline{Y}$? Thank you again. I understand completely this time. Mar 24 accepted If a chain of distinct irreducible closed subsets of a quasi-affine variety $Y$ is maximal, a chain of their closures is maximal in $\overline{Y}$? Mar 24 comment If a chain of distinct irreducible closed subsets of a quasi-affine variety $Y$ is maximal, a chain of their closures is maximal in $\overline{Y}$? Thank you for answering my question. I didn't notice that $W'$ is an open subset of W. One thing is not clear to me. You wrote $Z_i$ is not only irreducible in $Y$ but also in $X$. Does the irreducibility of a subset depend on the space containing it ? Mar 22 asked If a chain of distinct irreducible closed subsets of a quasi-affine variety $Y$ is maximal, a chain of their closures is maximal in $\overline{Y}$?