$p\in \mathbb Z$ is prime if and only if $\langle p\rangle$ is maximal.  Could someone give me a hint to prove it. please?
I know that if (p) is maximal then Z/(p) is a field which implies its an integral domain which then implies that (p) is a prime ideal and p is prime. I'm having trouble going the forward direction assuming that p is prime and then showing it's a prime ideal.
Error installing any module This applies to all d20 systems, but with an example from the from the Pathfinder  (emphasis mine):  Stonecunning: Dwarves receive a +2 bonus on Perception checks to potentially notice unusual stonework, such as traps and hidden doors located in stone walls or floors. They receive a check to notice such features whenever they pass within 10 feet of them, whether or not they are actively looking.  In this case, does the GM roll the check in secret or ask the player roll the check? If the GM asks the player to roll for the check, what stops situations like:  GM: (interrupting) &quot;Gimli, Roll a perception check.&quot; Dwarf player: &quot;Oh, it's a natural one, I guess I don't see anything.&quot; Human player: &quot;Okay, I'll roll a perception check for anything unusual in this room. Nice! 19. How about 19, does 19 work for you?&quot; GM: &quot;Gimli, everything looks kosher to you. Aragon, you notice a rock coloured tarp on the ground here, covering what looks like a pit trap.&quot;  In this case, by telling the dwarf to roll his &quot;free&quot; perception check, he's telegraphed to the players that there is something to notice. How can this be fixed, without bluffing checks for nothing in particular to throw off the signal to noise ratio of similar checks? Why does prime factorization for LCM work? A method to finding the LCM of two numbers is to use prime factoring. I know how to do this, but logically why does this work in finding the lcm Where did Harry's parents get all of their money from? Let $(A,+,\cdot)$ be a unity ring with the property that if $x \in A$ and $x^2=0$ then $x=0$. Consider the set $M=\{a\in A | a^3=a\}$. Prove that:      a) $2a\in Z(A)$, $\forall a\in M$, where $Z(A)$ denotes the centre of the ring $A$;   b) $ab=ba$, $\forall a,b\in M$.   My attempts revolved around the fact that an idempotent element in a reduced ring is central. So, since for $a\in M$ we have that $(a^2)^2=a^2$, it follows that $a^2\in Z(A)$, $\forall a\in M$. The next thing I wanted to use in order to solve a) was that $Z(A)$ is a subring of $A$, so if I had proved that $(a+1)^2 \in Z(A)$, $\forall a\in M$, then we would have reached the desired conclusion. However, I couldn't prove this and I honestly doubt that it is true. Another idea that I had was to prove that $M$ is a subring of $A$. Of course, this didn't work out because I cannot even prove that $M$ is closed under addition. Again, I don't know if this is true and it most likely isn't. As for b), I think that a) should be of use, but I don't know how. It is a well-known problem that a ring with $x^3=x$ for any $x$ in that ring is commutative, but since $(M,+,\cdot)$ is almost definitely not a ring, this doesn't help. EDIT: Is there any chance that this question is simply wrong? I tended to believe this before asking it here too, but since nobody has made any progress on it until now I am even more inclined to think so. Amy: What day is it today?    Tom: It's Monday. Hmm... I like Monday.   Amy: What do you have today?   Tom: We have math, science and art.What about you?   Amy: We have English and P.E. What do you have tomorrow?    Tom: We have art and P.E.tomorrow.    Amy: Oh, great.      I can understand that "What about you" means "What do you have today?"  But I wonder whether "What about you" is an elliptical sentence. If it is, what is the full sentence? If it isn't, how to parse this structure? Communicating with all potential clients as a multi niche photographer This may be seen as a bit off topic, but I'm hoping it's a question that some photographers can associalte with and clear up for me.  I'm an amateur wanting to get more serious in the new year, and aiming to build up a portfolio of work that I will showcase on my own website, which will hopefully build up a client base.  I figured that a good way to get started would be to create a few 365 style projects that I can publish on my site and on social media. However, my interests vary a great deal, and therefore my photography will more than likely range from nature and local events, to more NSFW style topics.  My problem is, how to successfully communicate with people who will be interested in certain aspects of my work, without losing touch with those interested in others.  In theory, I'd like to set up a single social media presence, preferably Twitter, where I can freely post anything I want, and connect with anyone I want. Essentailly pointing people in the direction of whatever content they will be interested in. Problem is, social media sites like Twitter publicise ALL of your posts to ALL of your followers, so people who are following for nature/local photography will possibly unfollow me for my NSFW content. At the same time, some SFW followers may be potential clients in other areas, and vice versa.  So as a multi niche photographer, how can you engage with all potential clients, and showcase all portfolio peices ( in my case all 365 project shots ) via a single or limited number of social networks? How are you doing it? Proof by induction that $2 + 4 + 6 + \cdots + 2n = n(n+1)$ How can I adjust the mouse scroll speed? How to deal a specific amount of damage using commands? I want to make a datapack where if you touch lime glass, you get damaged 2.5 hearts. I don't want the commands to work for just this specific value. Instead, I want to be able to specify the amount of damage. I can't use the instant damage effect because it's not controllable. byte[] to hex string How do I convert a byte[] to a string? Every time I attempt it, I get      System.Byte[]    instead of the value.  Also, how do I get the value in Hex instead of a decimal? Find the value of $\left[\frac{1}{\sqrt 2}+\frac{1}{ \sqrt 3}+......+\frac{1}{\sqrt {1000}}\right]$ Find the value of $$\left[\frac{1}{\sqrt 2}+\frac{1}{\sqrt 3}+.....+\frac{1}{\sqrt {1000}}\right]$$.Where [•] denote the greatest integer function.   I am very confused about this problem. I tried to find the upper and lower bound of the function. But I can't find any formulas to find the bounds. Somebody please help me. How many integer solutions does $xy+9(x+y)=2006$ have? Here $x$ and $y$ are both integers.  My trying:        I have tried to solve this problem. But I have no idea to solve this. Please help How to complain about a user? I  and a user was rude to me in a comment and in an answer. Even though I can't downvote, that answer got -5 and it was automatically deleted. I want to complain about that user. How can I do that? Aufbau principle in modern quantum theory Video editing software options? What video editing software would you recommend on Ubuntu (or Linux in general) and why?  It can be for either beginners or experienced users.  Please only place one piece of software per response and include details to what makes it great! How to completely delete and app from iphone (including all the data and history) So this problem particularly emerged with the Tinder app. I was using Tinder with a user account (A) and one day I logged out and created a new user account (B) which I never succeeded to activate due to sms verification problems, so I decided to login again with my old account. But even when I delete the Tinder app and reinstall it again, it still shows the B account logged in to the app. This means that IOS is somewhere storing a history for this app. Now I'm looking for a way to delete this history.  Note that this question is not necessarily about a certain app, but rather about where IOS stores historical data of apps and how to delete those histories alongside with the app.  P.S: If there are solutions like changing UUID of my i-phone that wouldn't damage my phone (I'm new to iphone so I'm not sure what I'm saying. For instance this jailbreaking thing which I keep hearing about and I'm not sure what it exactly does to my phone, doesn't sound like an I idea that I like), I'm open to them. Virtualbox Kernel driver not installed I tried to use my Virtual Box and this is the error I received - (I just updated to the new Ubuntu also) and I am VERY new at this Linux OS.     Kernel driver not installed (rc=-1908)      The VirtualBox Linux kernel driver (vboxdrv) is either not loaded or   there is a permission problem with /dev/vboxdrv. Please reinstall the   kernel module by executing      '/etc/init.d/vboxdrv setup'      as root. Users of Ubuntu, Fedora or Mandriva should install the DKMS   package first. This package keeps track of Linux kernel changes and   recompiles the vboxdrv kernel module if necessary. Which of these two is more correct and why?   The sun shines on my little tree. The sun shines upon my little tree.   From what I've read in dictionaries both are acceptable, however I would like to know if there are subtleties that might arise due to, for example, the context. Installing QGIS with ECW support on Mac?