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visits member for 2 years, 9 months
seen Jul 20 at 11:10

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 $\overline{Y}$?
Mar
7
awarded  Yearling
Mar
7
accepted Isomorphism of an extension field of a field of finite transcendence degree
Mar
7
asked Isomorphism of an extension field of a field of finite transcendence degree
Feb
5
answered If a neibourhood of the origin shrinks to the origin then its closure also shurinks to the origin?
Jan
31
accepted The kernel of homomorphism of a local ring into a field is its maximal ideal?
Jan
31
comment If a neibourhood of the origin shrinks to the origin then its closure also shurinks to the origin?
The definition in the book is as follows: $X$ is an F-space if its topology $\tau$ is induced by a complete invariant metric $d$.
Jan
31
asked The kernel of homomorphism of a local ring into a field is its maximal ideal?
Jan
31
asked If a neibourhood of the origin shrinks to the origin then its closure also shurinks to the origin?
Jan
21
comment The number of possible extensions of an embedding of a field into a algebraically closed field.
We can prove that $n=m$ by utilizing the isomorphism of $F(\alpha_1,\cdots,\alpha_k,\alpha_{k+1})[t]$ and $\sigma_{k+1}(F(\alpha_1,\cdots,\alpha_k,\alpha_{k+1}))[t]$, since the latter is a uniquely factrizable domain.
Jan
21
comment The number of possible extensions of an embedding of a field into a algebraically closed field.
After $\alpha_{k+1}$ a root of $f(t)$ is adjoined to $F(\alpha_1,\cdots,\alpha_k)$,and the extension $\sigma_{k+1}$ of $\sigma:F \rightarrow L$ to $F(\alpha_1,\cdots,\alpha_k,\alpha_{k+1})$ is obtained.$f(t)$ splits off a new factor $(x-\alpha_{k+1})^m$ in $F(\alpha_1,\cdots,\alpha_k,\alpha_{k+1})$. At the same time $\sigma_{k+1}f$ splits off a new factor $(x-\beta_{k+1})^n$ in $\sigma_{k+1}(F(\alpha_1,\cdots,\alpha_k,\alpha_{k+1}))$, where $\beta_{k+1}=\sigma_{k+1}(\alpha_{k+1})$. We can prove that $n=m$ (continues to next comment)