Jan
25
awarded  Enlightened
Jan
25
awarded  Nice Answer
Jan
11
comment Is $\phi: C^{\infty}(\mathbb{R}) \to (\mathbb{R},+), \phi(f) = f'(0)$ an isomorphism?
Thank you! ${}$
Jan
11
comment Is $\phi: C^{\infty}(\mathbb{R}) \to (\mathbb{R},+), \phi(f) = f'(0)$ an isomorphism?
Could you please give your questions better titles? This is your third question with the title "is this an isomorphism?".
Jan
10
comment Prove that factor rings are fields
Please search the site before posting. There already are many variations of this question.
Jan
10
comment Show that $(A / \mathfrak{a}) \otimes_A M \cong M / \mathfrak{a} M$ for a ring $A$, ideal $\mathfrak{a}$, $A$-module $M$.
It's a different presentation of the same argument. It might be more accessible to some. I actually added this to the comments when the question was posted. Since other proofs were posted, I didn't want my comments to be lost.
Jan
9
answered Show that $(A / \mathfrak{a}) \otimes_A M \cong M / \mathfrak{a} M$ for a ring $A$, ideal $\mathfrak{a}$, $A$-module $M$.
Jan
7
comment Show that $(A / \mathfrak{a}) \otimes_A M \cong M / \mathfrak{a} M$ for a ring $A$, ideal $\mathfrak{a}$, $A$-module $M$.
The isomorphism $M \cong A\otimes_A M$ and the exact sequence $\mathfrak a \otimes_A M \to A \otimes_A M \to A/\mathfrak a \otimes_A M \to 0$ when put together give the exact sequence $\mathfrak a \otimes_A M \xrightarrow{\varphi} M \xrightarrow{\psi} A/\mathfrak a \otimes_A M \to 0$. Since $\ker \psi = \im \varphi = \mathfrak aM$, the first isomorphism theorem gives the desired result.
Jan
7
comment Show that $(A / \mathfrak{a}) \otimes_A M \cong M / \mathfrak{a} M$ for a ring $A$, ideal $\mathfrak{a}$, $A$-module $M$.
I know you're looking for proof verification, but this proof is very long and technical. You can finish the proof quickly by using the isomorphism $M \cong M \otimes_A A$ and the exact sequence you have at the beginning.
Jan
3
comment Is the question in the Munkres's topology book wrong?
The definition of $[I, Y]$ does not require end points to be fixed, unlike the case with the fundamental group. This is a crucial difference.
Jan
2
comment Need help on how to compute the fundamental group of a space.
You can realize this space as a CW-complex by attaching one $2$-cell to a wedge sum of $5$ circles. The attaching map can be inferred from the diagram. This is one way to compute the fundamental group.
Dec
25
comment A question about the geometric representation of Spec $\Bbb{C}[x,y]/(x-y)$
It might be easier to see the correspondence via the isomorphism $\mathbb C[x, y]/(x-y) \cong \mathbb C[x]$.
Dec
21
comment Irreducible and prime elements
@BalarkaSen Not in general, no. See this.
Dec
20
comment Question related to the definition of affine schemes
No, the definition only requires the induced map on global sections to be an isomorphism. It does not have to be the identity.
Dec
20
comment Using the Maclaurin series to approximate $f(0.1)$ for $f(x)=(3+e^{2x})^{0.5}$
Please use MathJax to make the question readable. Help us help you.
Dec
19
awarded  Nice Answer
Dec
19
awarded  Constituent
Dec
18
reviewed Close How to compute the integral $\int^{\pi/2}_0\ln(1+\tan\theta)d\theta$?
Dec
18
answered Zero sets in completely regular spaces
Dec
18
comment Show that $\{e^{in}: n\in\Bbb N\}$ is Dense in the Unit Circle
Related question. The same technique can be used here.