Intereting Posts

Multiples of 4 as sum or difference of 2 squares
Concepts about limit: $\lim_{x\to \infty}(x-x)$ and $\lim_{x\to \infty}x-\lim_{x\to \infty}x$.
If $xy+xz+yz=3$ so $\sum\limits_{cyc}\left(x^2y+x^2z+2\sqrt{xyz(x^3+3x)}\right)\geq2xyz\sum\limits_{cyc}(x^2+2)$
Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$
Equivalence of geometric and algebraic definitions of conic sections
if $f$ is differentiable at a point $x$, is $f$ also necessary lipshitz-continuous at $x$?
Suppose that $G$ is a group with the property that for every choice of elements in $G$, $axb=cxd$ implies $ab=cd$. Prove that $G$ is Abelian.
Fourier Transform of $f(x) = \exp(-\pi ax^{2} + 2\pi ibx)$
Alligation or mixture
Prove that $A$ is diagonalizable iff $\mbox{tr} A\neq 0$
Proving that either $2^n-1 $ or $ 2^n+1$ is not prime
Euler-Maclaurin Summation
Prove Minkowski's inequality directly in finite dimensions
inequality using Lagrange Multipliers and Cauchy Schwarz inequality
How do I prove that a polynomial F of degree n has at most n roots

I was messing with the ring $k[x_1,\dots,x_n,\dots]$ of polynomials in numerable many variables in order to solve an exercise of Atiyah, and the following question came to me and made me curious:

Is there a commutative unitary ring $A$ isomorphic to $A[X]$ but not isomorphic to $A[x_1,\dots,x_n,\dots]$ the ring of polynomials with coefficients in $A$ and numerable many variables?

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- If a ring is Noetherian, then every subring is finitely generated?
- Left Invertible Elements of a monoid
- Are there any sets other than the usual in which we can apply Sturm's axioms?
- Braid invariants resource

- Is $R/N(R)$ a faithfully flat $R$-module?
- Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$
- Topics in combinatorial Group Theory
- For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
- Why principal ideals in Z are not maximal?
- How to find galois group? E.g. $\mathrm{Gal}(\mathbb Q(\sqrt 2, \sqrt 3 , \sqrt 5 )/\mathbb Q$
- Let $G=GL(2,\mathbb{Z}/5\mathbb{Z})$, the general linear group of $2 \times 2 $ matrices with entries from $\mathbb{Z}/5\mathbb{Z}$
- Prove every element of $G$ has finite order.
- Isomorphism between groups of real numbers
- For any rng $R$, can we attach a unity?

Let $\Gamma$ be the set of sequences $a_1 a_2 a_3 \cdots $ of nonnegative integers which are eventually constant. A typical element of $\Gamma$ looks like $3921726666666\cdots$. Note that $\Gamma$ is a commutative semigroup under addition. Let $A$ be the semigroup algebra $\mathbb{Z}[\Gamma]$. I claim that $A \cong A[t]$ but $A \not \cong A[t_1, t_2, t_3, \ldots]$.

The semigroup $\Gamma$ is isomorphic to $\mathbb{Z}_{\geq 0} \times \Gamma$, by the map $a_1 a_2 a_3 \cdots \mapsto (a_1, a_2 a_3 \cdots)$. So $\mathbb{Z}[\Gamma] \cong \mathbb{Z}[\Gamma \times \mathbb{Z}_{\geq 0}]$ or, in other words, $A \cong A[t]$.

Let $z$ denote the sequence $11111\cdots$ and let $x_i$ denote the sequence $000\cdots01000\cdots$ with the lone $1$ in the $i$th position. Note that $\Gamma$ embeds in the free abelian group generated by $z$ and the $x_i$, so $A$ embeds in the Laurent polynomial ring $\mathbb{Z}[z,x_1^{\pm},x_2^{\pm},x_3^{\pm},\ldots]$. In particular, $A$ is an integral domain. In this notation, the isomorphism $A \to A[t_1, t_2, \ldots, t_k]$ sends $z \mapsto z t_1 t_2 \cdots t_k$; it sends $x_i \mapsto t_i$ for $i \leq k$ and $x_i \mapsto x_{i-k}$ for $i >k$.

Now, suppose that $\phi$ is a map $A \to A[t_1, t_2, \ldots ]$ with $\phi(z)$ nonzero. We show that $\phi(A) \subseteq A[t_1, t_2, \ldots, t_N]$ for some $N$. In particular, $\phi$ is not an isomorphism.

Choose $N$ large enough that $\phi(z) \in A[t_1, t_2, \ldots, t_N]$. Since $\phi(x_i)$ divides $\phi(z)$, and $A$ is a domain, we must also have $\phi(x_i) \in A[t_1, \ldots, t_N]$. For any $\gamma$ in $\Gamma$, write $\exp(\gamma)$ for the corresponding element of $A$. For any $\gamma \in \Gamma$, we have $\prod_{i=1}^M x_i^{b_i} \exp(\gamma) = \prod_{i=1}^M x_i^{c_i} \cdot z^d$ for some sufficiently large $M$ and some $d$. Therefore, $\phi(\exp(\gamma))$ can be written as a ratio of $\prod_{i=1}^M \phi(x_i)^{c_i} \cdot \phi(z)^d$ and $\prod_{i=1}^M \phi(x_i)^{b_i}$, both of which are in $A[t_1, \ldots, t_N]$ (using again that $A$ is a domain). So $\phi(\exp(\gamma)) \in A[t_1, \ldots, t_N]$. Since the $\exp(\gamma)$ are a $\mathbb{Z}$ basis for $A$, this shows that $\phi(A) \subseteq A[t_1, \ldots, t_N]$.

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