In a ring with unity prove that if $a$ is nilpotent, then $a+1$ and $a-1$ are both invertible.    In a ring with unity we have: $1-a^n=(1-a)(1+a+a^2+...+a^{n-1})$. I believe this proves $a-1$ is invertible but I'm not sure why.  Then $1-(-1)^na^n=(1+a)(1-a+a^2-...+(-1)^{-1}a^{n-1})$. Again I think this proves it but I don't know how.  I apologize as we haven't discussed nilpotent rings and yet I was asked to answer this question.
Given a ring $R$ (with unity), prove that if $a \in R$ is nilpotent, then $a+1,a-1$ are both invertible.   Suppose $a^n=0$. Then $1=1-a^n=(1-a)(1+a+\ldots+a^{n-1})$, so $1-a$ is invertible.  If $n$ is odd we can use $1+a^n=(1+a)(1-a+\ldots+a^{n-1})$.  But what if $n$ is even?
Although I have installed the OpenLayers plugin in QGIS 2.4, I cannot see it in the list when I click on Plugins.    Does anyone else have this issue? What level is a spell if you cast it without expending a spell slot? Journals selection - alternative criteria to impact factor While trying to select the journals I can submit my present work and future work, I realize the time between the date the authors send the article to the journal and when the revised article is submitted. Usually this is something between 6 months and more than 12 months.  Althought I am focusing on the Impact Factor (IF) to select journals, I think that 12 months to have a feedback that might include "add more variables analysis","this article will be more valuable if you test also model x,y and z", can have a huge impact on you work. It will make you stop what you are currently doing and possible spend up to 2 months making the changes. This might make a big mess on your work if you have deadlines to respect, however this happens to everyone in academia.  My question is, besides the IF journal based selection, how can we select a journal with enought quality with lower IF? What alternative criteria should one use to select the journal? Should we choose newly created Journals? "export PS1=" for customizing shell prompt Show that if $p$ is prime then $\Bbb Z_p$ is a field Check my proof please.  Let $\Bbb Z_p:=\Bbb Z/p\Bbb Z$ be the quotient ring modulo $p$. I want to prove that if $p$ is prime then $\Bbb Z_p$ is a field.  Known facts about the quotient rings of the kind $\Bbb Z_n$ (I dont list all the characteristics of a ring, just some ones):   They are commutative rings with unity. $[0]=[n]$ is the identity of addition or zero. Then $[n]+[a]=[a]$ and $[n]\cdot[a]=[n]$ for all $[a]\in\Bbb Z_n$. $[1]$ is the multiplicative identity or unity. Then $[1]\cdot[a]=[a]$ for all $[a]\in\Bbb Z_n$. The addition is defined as $[a]+[b]=[a+b]$, and the multiplication is defined as $[a]\cdot[b]=[ab]$. $[a]=[b]$ means that exists $z\in\Bbb Z$ such that $a=b+nz$. $|\Bbb Z_p|=p$.   A ring $\Bbb Z_p$ dont have zero divisors, i.e. doesnt exist $[a],[b]\in\Bbb Z_p$ distinct of $[p]$ such that $[a]\cdot[b]=[p]$. Proof by contradiction: suppose that exists such $[a],[b]$ distinct of $[p]$ that are zero divisors. Then exist some $z_j\in\Bbb Z$ such that:  $$(a+pz_1)(b+pz_2)=pz_3\iff ab+pz_4=pz_3\iff ab=pz_5$$  but this is a contradiction with the assumption that $a,b\neq p$ and $p$ prime, so dont exist divisors of zero on any $\Bbb Z_p$.$\Box$  For any $[a]\in\Bbb Z_p$ distinct of $[p]$ exist $[b]$ such that $[a][b]=[1]$. Because $\Bbb Z_p$ dont have zero divisors and is finite for some $[a]\neq[p]$ we have that if $[b]\neq[c]$ then $[a][b]\neq[a][c]$. Proof by contradiction: if $[a][b]=[a][c]$ in the above conditions then  $$[a][c]=[ac]=[ab]=[a][b]\iff ac+pz_1=ab+pz_2\iff a(c-b)=pz_3$$  for some $z_j\in\Bbb Z$. But $pz_3\in[p]$, then $[a][c-b]=[p]$ but because there are no zero divisors in $\Bbb Z_p$ then or $[a]=[p]$ or $[c-b]=[p]$ what contradicts the above conditions.$\Box$  Because $[a][b]\neq[a][c]$ for $[a]\neq[p]$ and $[b]\neq[c]$ then for any $[a]\neq[p]$ we have that  $$[a]\cdot\Bbb Z_p=\Bbb Z_p\implies \exists [b]\in\Bbb Z_p:[a][b]=[1]$$  In other words: every element of $\Bbb Z_p$ but $[p]$ have a multiplicative inverse as stated above.$\Box$  Because $\Bbb Z_p$ is a commutative ring with $[1]\neq[0]$ and have multiplicative inverse for all their elements but $[p]=[0]$ then $\Bbb Z_p$ is a field.$\Box$ Why won't any faces show up in the UV Image Editor? Going into a directory linked by a link When I cd a link, my current path is prefixed with the link's path, rather than the path of the directory the link links to.  E.g.  ~/dirlinks/maths$ ls -l logic lrwxrwxrwx 1 tim tim 71 Jul 27 10:24 logic -&gt; /windows-d/academic discipline/study objects/areas/formal systems/logic  ~/dirlinks/maths$ cd logic ~/dirlinks/maths/logic$ pwd /home/tim/dirlinks/maths/logic  ~/dirlinks/maths/logic$ cd .. ~/dirlinks/maths$   I would like to have my current path changed to the path of the linked dir, so that I can work with the parent dirs of the linked dir as well. Besides ls the link to find out the linked dir, and then cd into it, what are some simpler ways to accomplish that? For example, after cd into a link, how do you change your current path to the path of the linked dir? When to use self references or composition? How can I vertically align elements in a div? I have a div with two images and an h1. All of them need to be vertically aligned within the div, next to each other. One of the images needs to be absolute positioned within the div. What is the CSS needed for this to work on all common browsers? &lt;div id=&quot;header&quot;&gt;   &lt;img src=&quot;..&quot; &gt;&lt;/img&gt;   &lt;h1&gt;testing...&lt;/h1&gt;   &lt;img src=&quot;...&quot;&gt;&lt;/img&gt; &lt;/div&gt; Table overflowing outside of div Could not load 'vboxdrv' after upgrade to Ubuntu 16.04 (and I want to keep secure boot) Induction problem: "if the group has at least one player who is better than Messi, then all the members of the group are better than Messi" How can I get this symbol without using bbold? Recently I commented on  question   and the text did got bold but the italic word didn't seem to get italic on Android Chrome(Latest Ver.), but when I saw from the PC browser, it rendered as it should...  I've attached a screenshot of my phone ... In .NET why is String.Empty read only instead of a constant? I'm just wondering if anyone knows what the reasoning was behind that decision. number of ways to make $2.00 How many different ways can you make $2.00 using only 1 cent, 5 cent, 10 cent, and 25 cent pieces, and 1 and 2 dollar bills (there are 100 cents in a dollar)? I have worked out an equation:  $$p + 5n + 10d + 25q + 100l + 200t = 200$$  Where all variables are non-negative integers. But I don't know what to do from here to figure out how many different sets of non-negative integers there are that satisfy this equation. What do I do next? Bump and color data in a single brush? Is there any way to paint onto a bump map and diffuse map at the same time, so that the strokes match up? Like a multi-layer brush that uses a greyscale texture for the bump layer and a color texture for the diffuse color?  Here's a rough sculpt to demonstrate what I'm imagining:    The GIF above shows how such a brush could be made. The black and white height map would be the displacement brush, transferring its greyscale data to the bump map texture and the green one would transfer its color data to the diffuse map texture.  The resulting brush images:      Making the brush is one thing, but is it possible to use such a brush? How does Router know where to forward packet If several computers with local addresses (192.168.0.#) are connected to a router and each computer opens a web browser and requests a page over HTTP, when these TCP:80 packets are sent out, the router switches the local address with the static IP of the router (i.e. Provider given IP) so the server can reply to the appropriate address.   But how does the router know to which computer to forward the HTTP reply, since the TCP header does not contain the local IP address (does it?), and all computers are using port 80?  Does this have anything to do with the MAC addresses?   How exactly does this work? I read something that suggests that if $I$ is the $n$-by-$n$ identity matrix, $v$ is an $n$-dimensional real column vector with $\|v\| = 1$ (standard Euclidean norm), and $t &gt; 0$, then $$\det(I + t v v^T) = 1 + t$$ Can anyone prove this or provide a reference? More generally, is there also an (easy) formula for calculating $\det(A + wv^T)$ for $v,w \in \mathbb{K}^{d \times 1}$ and some (invertible) matrix $A \in \Bbb{K}^{d \times d}$? Connecting to sql server database mdf file without installing sql server on client machine?