Intereting Posts

Find the principal value of $\int_{-\infty}^{\infty}\frac{1-\cos x}{x^2}\,\mathrm dx$
Evaluate $\int\frac{1}{1+x^6} \,dx$
Closed form for $\sum_{n=1}^\infty\frac{(-1)^n n^a H_n}{2^n}$
Prove that $g(x)=\frac{\ln(S_n (x))}{\ln(S_{n-1}(x))}$ is increasing in $x$, where $S_{n}(x)=\sum_{m=0}^{n}\frac{x^m}{m!}$
The binomial formula and the value of 0^0
Are there books introducing to Complex Analysis for people with algebraic background?
Prove the laws of exponents by induction
Axiomatic characterization of the rational numbers
Is there a constructive proof that a Euclidean domain is a UFD?
Why adjoining non-Archimedean element doesn't work as calculus foundation?
The Pythagorean theorem and Hilbert axioms
Steinitz exchange lemma
Showing Parallelism is an equivalence relation in $\Bbb R ^2$
Terms of a Sequence
Non-associative version of a group satisfying these identities: $(xy)y^{-1}=y^{-1}(yx)=x$

Let $R=k[x,y]$ be a polynomial ring ($k$, of course, is a field). Show that

$R/(xy-1)$ is not isomorphic to a polynomial ring in one variable.

- XOR is commutative, associative, and its own inverse. Are there any other such functions?
- Finite injective dimension of the residue field implies that the ring is regular
- Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
- What sort of algebraic structure describes the “tensor algebra” of tensors of mixed variance in differential geometry?
- Two exercises on characters on Marcus (Part 2)
- $X$ is a basis for free abelian group $A_{n}$ if and only if $\det (M) = \pm 1$
- number of non-isomorphic rings of order $135$
- Classifying groups of order 21.
- Good abstract algebra books for self study
- Understanding this proof that a polynomial is irreducible in $\mathbb{Q}$

Suppose $R\simeq K[T]$, where $K$ is a commutative ring. Then $K$ is an integral domain and since $\dim K[T]=1$ we get $\dim K=0$ (why?). It follows that $K$ is a field. So $k[X,X^{-1}]\simeq K[T]$. Now use this answer.

In the lines of the accepted answer, one can show that the global dimension of $k[x,x^{-1}]$ is $1$, so that if it is isomorphic to a polynomial ring, it must be isomorphic to one of the form $D[y]$ with $D$ a semisimple commutative domain, which is thus a field.

- Why is this not a triangulation of the torus?
- Evaluate the Sum $\sum_{i=0}^\infty \frac {i^N} {4^i}$
- Closed subspace of $l^\infty$
- Which of the (non-)Euclidean planes can we embed into non-Euclidean 3-space?
- On the number of possible solutions for a quadratic equation.
- Is there a nonnormal operator with spectrum strictly continuous?
- Primes and probabilities
- Calculate $\tan^2{\frac{\pi}{5}}+\tan^2{\frac{2\pi}{5}}$ without a calculator
- Are there infinitely many primes next to smooth numbers?
- Slick proof the determinant is an irreducible polynomial
- Why is sorting pancakes NP hard?
- A proof that $1=2$. May I know why it’s false?
- Proof of the inequality $2\uparrow^n 4 < 3\uparrow^n 3 < 2\uparrow^n 5$
- Explicit verification of signs in Morse complex
- how to determine if two graphs are not isomorphic