Show that the ring $A := \mathbb{C}[x, y]/(x^2 - y^2 - 1)$ is an integral domain. Further show that every nonzero prime ideal in A is maximal.  I proved that $A$ is an integral domain by showing that $x^2 - y^2 - 1$ is an irreducible in the UFD $\mathbb{C}[x, y]$ and hence a prime. To prove that every nonzero prime ideal in $A$ is maximal I need to prove either that:   i) A is a PID, so in a sense I need to be able to express $x$ in terms of $y$ (or vice versa) but I only know that $x^2 = y^2 + 1$ in $A$.  ii)If $I$ is a prime ideal in $A$ then every non zero element in $A/I$ is invertible.
Let $R=\frac{\mathbb{C}[x,y]}{\langle x^2-y^2-1 \rangle}$. Prove that (1) $R$ is an integral domain. (2) Any nonzero prime ideal of $R$ is maximal.  My idea: (1) The first part is easy. Since $x^2-y^2-1\in \mathbb{C}[y][x]$ is Eisenstein with respect to the prime $y+i$, it is irreducible. Hence the ideal generated by it is a prime ideal. (2) I am struggling with this part. I have one approach (not sure if correct). Approach : Let $I=\langle x^2-y^2-1 \rangle$. Then $R=\mathbb{C}[x,y]/I$.  Any ideal of $R$ is of the form $J/I$ where $J\subseteq \mathbb{C}[x,y]$ is an ideal containing $I$. If we can show that any prime ideal $J$ containing $I$ is maximal, then we are done. How to prove this? Is there any other way to prove the second part? Edit: I am trying to avoid Hilbert Nullstellensatz and the notion of dimension of a ring, as these concepts are not there in the syllabus of the exam from which I found this question.
I used to use a lot ctrl+alt+numpad to move windows, how can I use this feature on ubuntu 18 like it was default before?  Thank you in advance Java Reflection Performance Does creating an object using reflection rather than calling the class constructor result in any significant performance differences? I cannot remember anyone writing the letter epsilon in any other way than \varepsilon in any math class; but in LaTeX \epsilon and \varepsilon are different symbols. Do any of you know why there are two different symbols? (I.e. if \epsilon is the correct way to write the letter epsilon, why aren't mathematicians using it, and when is, according to the standards today, the correct situation to use each of the symbols?) Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$ NULL in MySQL (Performance & Storage) What exactly does null do performance and storage (space) wise in MySQL?  For example:  TINYINT: 1 Byte TINYINT w/NULL 1 byte + somehow stores NULL? Project Euler - 34 / Find a mathematical approach for upper bound 145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145. Find   the sum of all numbers which are equal to the sum of the factorial of   their digits. Note: as 1! = 1 and 2! = 2 are not sums they are not   included.   So I tried a random enough large upper bound (100000) in my program and it worked.  I deleted this part to not spoil the answer  My question is : How to find the closest upper bound using a mathematical approach ? Can I ask a postdoc who has closely supervised my research to write me a letter of recommendation? "None is" or "none are" I was asked if the following sentences will create problems and what are the assumptions I have to make:  (d) There are grammarians in the room but none of them is an American. (e) There are grammarians in the room but none of them are Americans.  What is the problem in both sentences? They clearly have two main clauses which are connected by the coordinated conjunction 'but'. I know that grammar-wise, the 'is/are' don't really matter since both can be used with 'none', but would the tense in the first main clause 'are' affect the use of 'none'? How to extract the n-th elements from a list of tuples? Is it advisable to use lead-free PCB and solder paste on a first SMD assembly job? What are bitwise shift (bit-shift) operators and how do they work? I've been attempting to learn C in my spare time, and other languages (C#, Java, etc.) have the same concept (and often the same operators) ...   What I'm wondering is, at a core level, what does bit-shifting (&lt;&lt;, &gt;&gt;, &gt;&gt;&gt;) do, what problems can it help solve, and what gotchas lurk around the bend? In other words, an absolute beginner's guide to bit shifting in all its goodness. What happens when you need to tear-down a hotel but no houses are in the bank in Monopoly? How can I manually create a pixel perfect image sprite? $$\lim_{n \to\infty} \frac{n!}{n^n}$$  I have a question: is it valid to use Stirling's Formula to prove convergence of the sequence? An example to show that convergence of Cesaro sum to $0$ does not imply the original sequence converges to $0$. I'd like construct a non-negative sequence $\{a_{k}\}_{k=0}^{\infty}$ with $a_{0}=0$ such that the Cesaro sum $\frac{1}{n}\sum_{k=0}^{n-1}a_{k}\longrightarrow 0$ but $a_{n}$ does not converge to $0$.   I have a really bad attempt:  Define $a_{0}=a_{1}=0$, and $a_{k}:=\log(\log(k))$ for $k\geq 2$, then $a_{k}\longrightarrow\infty$. But $$\dfrac{1}{n}\sum_{k=0}^{n-1}a_{k}=\dfrac{\log\Big(\prod_{k=2}^{n-1}\log(k)\Big)}{n}$$ and Wolframalpha told me that this series converges to $0$ as $n\longrightarrow\infty$.  However, I have no idea about how to show this convergence to $0$. Also, I wish to have a sequence as simple as possible, since at some stage I need too how $f(k):=a_{k}$ is positive semi-definite.   Is there any simpler example?   Thank you! Ephemeral error message: "cryptsetup: unknown fstype, bad password or options?" Should I put a comma before the last item in a list? Should I put a comma before the last item in a list?     I would like crackers, cheese and some soda.   I would like crackers, cheese, and some soda. Limit of sequence in which each term is defined by the average of preceding two terms We have a sequence of numbers $x_n$ determined by the equality   $$x_n = \frac{x_{n-1} + x_{n-2}}{2}$$  The first and zeroth term are $x_1$ and $x_0$.The following limit must be expressed in terms of $x_0$ and $x_1$ $$\lim_{n\rightarrow\infty} x_n $$  The options are:   A)$\frac{x_0 + 2x_1}{3}$ B)$\frac{2x_0 + 2x_1}{3}$  C)$\frac{2x_0 + 3x_1}{3}$ D)$\frac{2x_0 - 3x_1}{3}$  Since it was a multiple choice exam I plugged $x_0=1$ and $x_1=1$. Which means that all terms of this sequence is $1$,i.e, $$x_n=1, n\in \mathbb{N} $$  From this I concluded that option A was correct.I could not find any way to solve this one hence I resorted to this trick. What is the actual method to find the sequence's limit? I'm curious why PCBs have one square pad in row for 0.1" headers. What is purpose of putting one square pad? What it indicates? Can you push more amps through square pad than round? Would the latter be an issue with the funding body, when the project proposal and budget only covers the expenses for hiring one postdoc? Note: it is the supervisors who got the grants, not the postdocs themselves.