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### A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
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### The Ring Game on $K[x,y,z]$

I recently read about the Ring Game on MathOverflow, and have been trying to determine winning strategies for each player on various rings. The game has two players and begins with a commutative ...
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### Does there exist a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?

Let $(X,\tau), (Y,\sigma)$ be two topological spaces. We say that a map $f: \mathcal{P}(X)\to \mathcal{P}(Y)$ between their power sets is connected if for every $S\subset X$ connected, $f(S)\subset Y$ ...
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Let $$H=\left(\begin{array}{cccc} 0 & 1/2 & 0 & 1/2 \\ 1/2 & 0 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2\\ 0 & 1/2 & 1/2 & 0 \end{array}\right),$$ $K_1=\left(\... 0answers 3k views ### Is there a categorical definition of submetry? (Updated to include effective epimorphism.) This question is prompted by the recent discussion of why analysts don't use category theory. It demonstrates what happens when an analyst tries to use ... 0answers 1k views ### What functions can be made continuous by “mixing up their domain”? Definition. A function$f:\Bbb R\to\Bbb R$will be called potentially continuous if there is a bijection$\phi:\Bbb R\to\Bbb R$so that$f\circ \phi$is continuous. So one could say a potentially ... 0answers 6k views ### Is this really a categorical approach to integration? Here's an article by Reinhard Börger I found recently whose title and content, prima facie, seem quite exciting to me, given my misadventures lately (like this and this); it's called, "A Categorical ... 0answers 3k views ### Is there a characterization of groups with the property$\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$? A common mistake for beginning group theory students is the belief that a quotient of a group$G$is necessarily isomorphic to a subgroup of$G$. Is there a characterization of the groups in which ... 0answers 2k views ### Application of Hilbert's basis theorem in representation theory In Smalo: Degenerations of Representations of Associative Algebras, Milan J. Math., 2008 there is an application of Hilbert's basis theorem that I don't understand: Two orders are defined on the set ... 0answers 2k views ### What is the Picard group of$z^3=y(y^2-x^2)(x-1)$? I'm actually doing much more with this affine surface than just looking for the Picard group. I have already proved many things about this surface, and have many more things to look at it, but the ... 0answers 1k views ### Does every ring of integers sit inside a ring of integers that has a power basis? Given a finite extension of the rationals,$K$, we know that$K=\mathbb{Q}[\alpha]$by the primitive element theorem, so every$x \in K$has the form $$x = a_0 + a_1 \alpha + \cdots + a_n \alpha^n,$$ ... 1answer 1k views ### Ring structure on the Galois group of a finite field Let$F$be a finite field. There is an isomorphism of topological groups$\left(\mathrm{Gal}(\overline{F}/F),\circ\right) \cong (\widehat{\mathbb{Z}},+)$. It follows that the Galois group carries the ... 0answers 4k views ### Pullback and Pushforward Isomorphism of Sheaves Suppose we have two schemes$X, Y$and a map$f\colon X\to Y$. Then we know that$\operatorname{Hom}_X(f^*\mathcal{G}, \mathcal{F})\simeq \operatorname{Hom}_Y(\mathcal{G}, f_*\mathcal{F})$, where$\...
Let $g_i$ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $(g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence ...