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Jul
4
comment every $F$-algebraic homomorphism of $K$ is $1-1$ and onto?
As indicated by comments on that question, $\mathbb Q$ can be replaced by any field $F$. The set of roots $S$ will still be finite.
Jul
4
comment Is it true that $M\otimes_A F\simeq M^{(I)}$?
The tensor product functor is left adjoint. Hence, it commutes with colimits.
Jul
4
comment Is the product topology the most finest topology you can give to the cartesian product and why?
Then any topology that is finer than the product topology would do. If you want uniqueness, you need to appeal to a universal property.
Jul
1
comment Trying to use the Zariski topology in a problem without knowing scheme theory.
@pseudoname123456 $A$ is a finitely generated algebra over a field, so it is Jacobson. For what it's worth, my argument doesn't require $I$ to be radical either.
Jun
30
comment Trying to use the Zariski topology in a problem without knowing scheme theory.
It's unclear to me what you're asking. Points in $S$ are closed. It follows that $S$ is discrete and hence zero-dimensional. We conclude that $A$ is zero-dimensional as desired. What's crucial here is recognizing the correspondence between irreducible subsets and prime ideals. In particular, there is a correspondence between points and maximal ideals. This follows from the Nullstellensatz.
Jun
23
revised Inclusion in cone is homotopy equivalence
Fix \sim spacing
Jun
19
revised Find the Distance Point to Line with Point on Line and Direction Vector
edited tags
Jun
17
revised Compute the area of specific shapes
edited tags
Jun
14
revised Zero set of a homogeneous element of degree $0$, or how $D_+(2)\subset \text{Proj}(\mathbb{Z}[x])$ looks like.
Fixed my name
Jun
14
comment Zero set of a homogeneous element of degree $0$, or how $D_+(2)\subset \text{Proj}(\mathbb{Z}[x])$ looks like.
Why isn't $(p, f(x))$ homogeneous? If $f(x)$ is homogeneous, then $(p, f(x))$ is generated by homogeneous elements. They don't have to be of the same degree.
Jun
10
comment $z\exp(z)$ surjectivity with the Little Picard Theorem
@TheSubstitute Suppose $-w = \exp(g(w'))$. Then $g$ misses almost all of $g(w') + n(2\pi i)$.
Jun
9
comment Inclusion induces identity on homology
$\mathbb Z / n \mathbb Z$ is a subgroup of $\mathbb Z^2$ if $n = 0$ too. But this would imply $\mathbb Z = 0$ by the first part of the sequence.
Jun
6
comment Exercise $1.8$ of chapter one in Hartshorne.
@user26857 Indeed. What's left now is a straightforward application of Krull’s Hauptidealsatz.
Jun
6
answered Exercise $1.8$ of chapter one in Hartshorne.
Jun
6
revised What is the need to define so many forms of equation of a straight line?
edited tags
Jun
1
comment For what kind of $R$-modules $M$ can we find an element $m\in M$ satisfing that $i:M\to M\otimes_R M, x\mapsto x\otimes m$ is an epimorphism?
@Bernard $R = \mathbb Z$, $M = \mathbb Z / (2)$ and $m = 1$.
May
31
comment Mayer-Vietoris in reduced homology for a torus.
Dear @Far, please edit your question with this information.
May
31
comment Noetherian ring under some conditions has at least two minimal prime ideals
Dear @Praphulla, I understand this. I just wanted my answer to be self-contained.
May
31
revised Noetherian ring under some conditions has at least two minimal prime ideals
added 165 characters in body
May
31
comment Noetherian ring under some conditions has at least two minimal prime ideals
Not to be pedantic, but the zero ring is commutative with $1$. It just happens that $0 = 1$. I don't have the book next to me. I guess it assumes $0 \ne 1$ at the beginning of the chapter.