Let R be a commutative ring with identity, and I,J its two ideals. One book says that $Spec (R/I \otimes R/J)=Spec(R/(I+J)) $. But I just cannot verify it. Is $R/I \otimes R/J=R/(I+J) $ right? I cannot get it. Or is there some other way to prove this result?
Let $R$ be a commutative ring with identity and $I$ and $J$ be ideals in $R$.     Show that $R/I\otimes_R R/J\cong R/(I+J)$ as $R$-modules.   This is what I tried.  Define a map $f:R/I\times R/J\to R/(I+J)$ as $f(\bar r, \bar s)=rs+(I+J)$. This is a well-defined bilinear map.  By the universal property of tensor products, we get an $R$-linear map $F:R/I\otimes_R R/J\to R/(I+J)$ which sends $\bar r\otimes \bar s$ to $rs+(I+J)$.  This map is surjective.  So all we need to show is that the kernel of $F$ is $0$.  But I am unable to do this.
Number of Pairs of Subsets Find the number of Pairs $(A,B)$ of subsets of $[n]$ such that $A ⊆ B$? I just need clarification in my thought process. My professor's wording at times can through me off. I just want to know if I am on the right track. We will have $3$ regions: 1.Outside $B$ 2.Inside $A$ 3. Inside $B$, Outside $A$ I believe the number of pairs is $3^n$ but I am not sure. Any help would be appreciated. While executing an INSERT statement with many rows, I want to skip duplicate entries that would otherwise cause failure. After some research, my options appear to be the use of either:  ON DUPLICATE KEY UPDATE which implies an unnecessary update at some cost, or INSERT IGNORE implies an invitation for other kinds of failure to slip in unannounced.  Am I right in these assumptions? What's the best way to simply skip the rows that might cause duplicates and just continue on to the other rows? When should you push variables in quotes in boolean conditions? Are the following two boolean expressions the same?  if [ -n $1 ] ; then  if [ -n "$1" ] ; then   If not - When should you put a variable in quotes? Inequality Proof in Real Numbers: If $p+q=1$ then $pq\le\frac14$ Let $p,q \in \mathbb R$, and $p+q=1$. Prove that $$pq \le \frac{1}{4}.$$   The first thing I did was define 3 possibilities that we can have from $p+q=1$:  1º  $p$ or $q$ is negative. Example:   $p=-10$, $q=11$   $-10+11=1$  $(-10)(11)=-110$  So if $p$ or $q$ are negative the inequality $pq$ $\le$ $\frac14$ holds.  2º $p$ or $q$ is $0$, that makes the inequality $pq$ $\le$ $\frac14$ be $0$ $\le$ $\frac14$, the inequality holds.  3º The final possibility I saw  was $0$ $\lt$ $p,q$ $\lt 1$, here i just see that the max value from $p,q$ can be $p,q = 1/2$ making $\frac12 + \frac12 = 1$ and $(\frac12) (\frac12) \le \frac14$, but i do not see the elegant way of proving that with other values the inequality holds.  I apologize in advance if I made any mistake or missed something. How do I remove repeated elements from ArrayList? I know this may be a tough or confusing question, but I have numerous existing materials I created and reuse for a project, which were all prepared for Blender Internal.  Now, I want to start the transition to Cycles, but I've hit a roadblock. How can I transition my materials without completely starting over? Or should I just forget it and start with fresh materials? Typically Apt packages are served over bare HTTP and the downloaded contents are verified with GPG signatures. This makes it easy for  to proxy the HTTP connections and add the downloaded packages to its cache. apt-cacher-ng is unable to directly MITM an HTTPS connection created by Apt. It is possible to  apt-cacher-ng, but then the contents are not cached--they are re-downloaded from the remote server every time. This quickly gets annoying when rebuilding Docker containers that install many packages from Apt repositories served over HTTPS. Is there a way to get apt-cacher-ng to cache packages downloaded from HTTPS repositories? I have read both two uses:   The combinations of a, b and c, taken two at a time, are ab, bc, ac. We can not help but noticing a harmonious combination between classic and contemporary.   Are both uses interchangeable? Perhaps substitute combinations among for combinations of in the first sentence? Any idea is appreciated. What is Active Directory Domain Services and how does it work? This is a  about Active Directory Domain Services (AD DS).         I find myself explaining some of what I assume is common knowledge about it almost daily. This question will, hopefully, serve as a canonical question and answer for most basic Active Directory questions. If you feel that you can improve the answer to this question, please edit away. Why do soap operas have the soap opera effect? The soap opera effect occurs when frame interpolation is used. For example, if a movie is shot at 24 frames per second and then shown on a TV screen which is capable of displaying 60 frames per second then it is possible to use a computer to do motion interpolation. In other words the computer can average (interpolate) two frames and create a synthetic third frame and stick it between the two real frames.  This interpolation makes the picture look unnatural and plastic, somewhat like soap operas look on TV, hence the name.  My understanding of why soap operas have the soap opera effect has something to do with the fact that many of them are shot using camcorders instead of film, but I am not completely sure.  Why do soap operas have the soap opera effect? $R/Ra$ is an injective module over itself Let $R$ be a PID, $a\in R$ be a nonzero nonunit in $R$. Prove that $R/Ra$ is an injective module over itself.   If $R$ is a PID, every $R$- divisible module is injective, but the question concerns with $R/Ra$-module, so I have no idea to solve this problem. Help me. Finding the confidence level and number of successes? Given that the sample size is $1500$ and the confidence interval of $p$ is $(0.050, 0.083)$. Find the confidence level and the number of successes.  I have so far that $E=Z_{1-\alpha/2}∗\sqrt{(p∗({p-1}))/n}$   Where $p$ is the center of the given interval, so $p = (0.050+0.083)/2 = 0.0665$  Also, the error $E$ as the difference of the midpoint and the endpoint of the interval so $E=0.0665-0.050=0.0165$   Plugging this into the equation I got that  $0.0165=Z_{1-\alpha/2}*\sqrt{((0.0665)(1-0.0665))/1500}$   Solving for $Z_{1-\alpha/2}$ gives $Z_{1-\alpha/2}=2.56$ and since $Z_{2.96} = 0.99$   $Z_{1-\alpha/2} = 0.01$  Is what I have so far right? If so, how do I find the number of successes? Is it just to multiply 0.99 by the number of trials?   So that way I would get $1500*0.99=1485$ ? Sensor for ultralow (to -85°C) freezer monitor The problem:  Consider the smooth quadric $Q=V(X_{0}X_{1}+X_{2}X_{3}+X_{4}^{2})\subset\mathbb{P}^{4}$ and the line $L=V(X_{0},X_{2},X_{4})$ contained in it. Prove that there exists a vector bundle $F$ on $Q$ with a section vanishing exactly at $L$.  The hardest thing is to get the transition matrices. Any help is appreciated. Can I get a postdoc in China more than 3 years after having received my PhD? Sending packets captured with Wireshark I am unable to find the link for downloading Ubuntu 16.04 onto my   desktop/laptop. I am able to see the link only for 18.04, but I need 16.04. Book with a girl whose grandma is a phoenix, cover depicts the emerald/green-eyed blonde girl I read this book in 2016, but I remember the cover being torn up, so it couldn't have been a new book. There was a girl's face on the cover. She had lightly colored skin, and beautiful emerald green eyes. Her hair was a dirty blond. This book was first in a trilogy and the cover was illustrated.  The girl's grandmother in the story was actually a phoenix (I think this was revealed in a different book in the trilogy though). The book started out in a small village, and I can't remember exactly where in the village chapters (the chapters that took place in the village) but she sneaked her deceased mother's green dress out of a trunk she wasn't supposed to go in, and I think she wore it to some event. UITableView cells strangely disappearing How to reduce title bar height?