I am currently learning abstract algebra, and I am going over prime elements of a ring. I am a little confused as to the connection between the definitions of prime numbers and prime elements. A prime number is an integer $p$ greater than $1$ such that $a\mid p \Rightarrow (a=1 \vee a=p)$ for an integer $a$ greater than $0$. A prime element is an element $p$ of a commutative ring with identity such that for $x,y\in R$, $p\mid xy \Rightarrow (p\mid x \vee p\mid y)$. Now, I know that a prime element is more general than a prime number. I just want to prove that if $p$ is a prime number in the integers, then it satisfies the prime element property. Can anyone provide a proof? I'm completely stuck.
The question is:      Prove that if $p \in \mathbb{Z}$ is irreducible, then $p$ is also prime.    Irreducible is defined as follows: $n \in \mathbb{Z} $ is irreducible iff the only way to write $n=ab$ for some integers $a$ and $b$ is if $a = \pm1$ or $b = \pm1$. Prime is defined as follows: an integer $n$ is prime iff whenever $a,b \in \mathbb{Z} $ and $p\mid ab$ then it must be true that $p\mid a$ or $p\mid b$ (or both).    Hint: Prove that if $n\mid ab$ but $n$ does not divide $a$, then $\gcd(a,n) = \pm1$. Then use Bézout's theorem and prove $n\mid b$.
How to calculate the number of days you can stay in Taiwan on a tourist visa? Tourist visas to Taiwan have an associated "maximum stay" of 30 days. For those visas, how are the days counted? Does the stay begin on the partial day that the traveler arrives or the first full day spent in the country? Does it end on the last full day spent in the country, or the actual day the traveler departs? What reputation do you need on Stack Overflow to receive an invite to Careers? Example of non-flat modules Let $R = \mathbb{C}[t]$ be a ring of polynomials in variable $t$ with coefficients in the field of complex numbers $\mathbb{C}$ and let $$N = R[x]/(tx-t).$$  I claim that $N$ is not a flat $R$- module. If we consider the exact sequence $$0 \rightarrow(t) \rightarrow R$$ such that the ideal $(t)$ is viewed as an $R$-module.   We know that the above sequence exact iff the map $(t) \rightarrow R$ is injective. Let us assume that N is flat then this means that $N \otimes  (t) \rightarrow N \otimes R$ has to be injective. How do I go from here?   Thanks. Consider the sentence      E-books are on the rise, but they haven't suppressed paper books though.    This usage seems to be quite common, but when I learned English  I was taught to use "however" where I now read "though". My questions on this use of "though" are:     Is this a new trend? Is it restricted to American English? Is "though"  considered more colloquial than "however" ?    And on a syntactic level: Do you put a comma before "though"? Humanity meeting another similar intelligent life form deep into space (or not)? I'm looking for a plot line (it should be a short story) that depicts Earth population (or their close descendants) establishing contact with another intelligent life form deep into space.   After careful studying of DNA samples and language / communication symbols they then realize that the civilization in question is also from Earth but that they departed very long ago and travelled throughout the galaxy in the completely opposite direction so that they are now meeting practically at the point of the circle opposite to Earth. There might have been some anatomical differences as well.  As far as I can remember, the story concludes shortly after that.  This should be a work of a well-known SF author, so I'm hoping for a quick response (if time didn't corrupt my memory too much).  I don't think it was part of an anthology, but it's entirely possible it was in a magazine.  As for the timeline, I've read it about 2-3 years ago, but since I have a passion for older SF works it might not be of much use.  Edit: Not sure if this could help but for what it's worth, it might be contained in . Is phonetic typing for Tamil available in Ubuntu 12.04?  How can I install it? What are the norms in Ito isometry? Itō isometry from :     Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical   real-valued Wiener process defined up to time $T &gt; 0$, and let $X :  [0, T] \times \Omega \to \mathbb{R}$ be a stochastic process that is   adapted to the natural filtration $\mathcal{F}_{*}^{W}$ of the Wiener   process. Then $$      \mathbb{E} \left[ \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \right)^{2} \right] = \mathbb{E} \left[ \int_{0}^{T} X_{t}^{2} \,  \mathrm{d} t \right], $$ where $\mathbb{E}$ denotes expectation with   respect to  $\gamma$. In other words, the Itō   stochastic integral, as a function, is an isometry of normed vector   spaces with respect to the norms induced by the inner products $$      ( X, Y )_{L^{2} (W)} := \mathbb{E} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t} \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) =  \int_{\Omega} \left( \int_{0}^{T} X_{t} \, \mathrm{d} W_{t}  \int_{0}^{T} Y_{t} \, \mathrm{d} W_{t} \right) \, \mathrm{d} \gamma  (\omega) $$ and $$      ( A, B )_{L^{2} (\Omega)} := \mathbb{E} ( A B ) = \int_{\Omega} A(\omega) B(\omega) \, \mathrm{d} \gamma (\omega). $$   I was wondering what the two normed spaces are and what their norms are, so that the two normed spaces are isometric wrt their norms?  Thanks and regards! Connecting desktop computer to the internet through laptop wifi Due to some home network complications I have had to move my router to a seperate part of the house, therefore the wired network I had set up can no longer work. Before I find the time to go out and buy a Wifi adapter for my desktop PC, I have a laptop that uses a built in Wifi card to connect to my router and this can connect to the internet, and I was wondering if I could somehow access the internet on my desktop PC via my laptop.  I'm hoping for a not-so-complex solution as this will only be set up for a few days, but it is quite vital that my desktop computer gets internet access.  Does anyone have experience in this sort of thing and can help me out? Thanks. How can I find out if Ubuntu supports a particular hardware component, that is, if it works in Ubuntu?  I am going to put together a computer and I am choosing components.   How do I know if Ubuntu supports this or that motherboard, or this or that sound card, or this or that graphics card, etc? How do I know that there is no conflict in Ubuntu between this and that hardware component?   What resources are there to help with such situations?  I know that there are computers with Ubuntu pre-installed and that there are  for whole computers but this is not what I'm looking for. I'm looking for resources to find out support for particular hardware components and conflicts among particular hardware components. onKeyUp not firing for ESC in IE Real Time LaTeX? Possible Duplicate:          I'm looking for an application in which I can write pure LaTeX in one half of the screen and on the other half to see the compiled document in real time. This seems really like an obvious way to do LaTeX authoring but it doesn't look like any software is able to do it. LyX is not what I'm looking for. I want to edit the actual LaTeX and see it rendered in real time. Why isn't this already being done somewhere? Visiting two Schengen countries, coming back home in between, do I apply for single-entry or multiple-entry visa? so i change my MAC address with macchanger -A wlp68s0b1 at boot with crontab, here is what happens when i disconnect and reconnect :  while connecting after boot :  rahman@debian:~$ macchanger -s wlp68s0b1 Current MAC:   00:22:31:c6:38:45 (SMT&amp;C Co., Ltd.) Permanent MAC: 00:00:00:00:00:00 (FAKE CORPORATION)   after disconnecting :  rahman@debian:~$ macchanger -s wlp68s0b1 Current MAC:   16:7b:e7:3c:d3:cd (unknown) Permanent MAC: 00:00:00:00:00:00 (FAKE CORPORATION)   after reconnecting :  rahman@debian:~$ macchanger -s wlp68s0b1 Current MAC:   00:00:00:00:00:00 (FAKE CORPORATION) Permanent MAC: 00:00:00:00:00:00 (FAKE CORPORATION)   and so on, and with every disconnect i get a different random MAC address which fades on reconnecting giving me my real MAC address , what causes that and how to stop it ?  some outputs :  rahman@debian:~$ lspci -nn |grep 14e4 44:00.0 Network controller [0280]: Broadcom Limited BCM4313 802.11bgn Wireless Network Adapter [14e4:4727] (rev 01) rahman@debian:~$ uname -a Linux debian 4.9.0-3-amd64 #1 SMP Debian 4.9.30-2+deb9u5 (2017-09-19) x86_64 GNU/Linux rahman@debian:~$ sudo ifconfig  enp0s25: flags=4099&lt;UP,BROADCAST,MULTICAST&gt;  mtu 1500         ether 00:24:c0:7b:a8:8b  txqueuelen 1000  (Ethernet)         RX packets 0  bytes 0 (0.0 B)         RX errors 0  dropped 0  overruns 0  frame 0         TX packets 0  bytes 0 (0.0 B)         TX errors 0  dropped 0 overruns 0  carrier 0  collisions 0         device interrupt 20  memory 0xd4800000-d4820000    enp0s25:avahi: flags=4099&lt;UP,BROADCAST,MULTICAST&gt;  mtu 1500         inet 169.254.9.109  netmask 255.255.0.0  broadcast 169.254.255.255         ether 00:24:c0:7b:a8:8b  txqueuelen 1000  (Ethernet)         device interrupt 20  memory 0xd4800000-d4820000    lo: flags=73&lt;UP,LOOPBACK,RUNNING&gt;  mtu 65536         inet 127.0.0.1  netmask 255.0.0.0         inet6 ::1  prefixlen 128  scopeid 0x10&lt;host&gt;         loop  txqueuelen 1  (Local Loopback)         RX packets 9436  bytes 6584515 (6.2 MiB)         RX errors 0  dropped 0  overruns 0  frame 0         TX packets 9436  bytes 6584515 (6.2 MiB)         TX errors 0  dropped 0 overruns 0  carrier 0  collisions 0  wlp68s0b1: flags=4163&lt;UP,BROADCAST,RUNNING,MULTICAST&gt;  mtu 1500         inet 192.168.1.5  netmask 255.255.255.0  broadcast 192.168.1.255         inet6 fe80::6711:9875:eb78:24fc  prefixlen 64  scopeid 0x20&lt;link&gt;         inet6 fd9c:c172:b03b:ce00:f1e0:695e:7da0:91a  prefixlen 64  scopeid 0x0&lt;global&gt;         ether 00:00:00:00:00:00  txqueuelen 1000  (Ethernet)         RX packets 484346  bytes 641850809 (612.1 MiB)         RX errors 0  dropped 0  overruns 0  frame 0         TX packets 368394  bytes 44259668 (42.2 MiB)         TX errors 0  dropped 0 overruns 0  carrier 0  collisions 0  rahman@debian:~$ sudo iwconfig  lo        no wireless extensions.  enp0s25   no wireless extensions.  wlp68s0b1  IEEE 802.11  ESSID:"3bdo"             Mode:Managed  Frequency:2.447 GHz  Access Point: 9C:C1:72:B0:3B:D4              Bit Rate=65 Mb/s   Tx-Power=30 dBm              Retry short limit:7   RTS thr:off   Fragment thr:off           Encryption key:off           Power Management:off           Link Quality=54/70  Signal level=-56 dBm             Rx invalid nwid:0  Rx invalid crypt:0  Rx invalid frag:0           Tx excessive retries:4  Invalid misc:183   Missed beacon:0 In Avatar: The Last Airbender, Aang's eyes glow when he goes into the Avatar State. This seems to be the case with all Avatars, but Aang and the other airbending Avatars tattoos glow too.      Is there any reason for why airbending Avatars tattoos glow as well as their eyes? Let $X,Y$ be topological and $f:X \to Y$ be open and continuos.  I am studying Baire space and I would like to try the following facts :      $(i)$ The inverse image of dense set is dense.       $(ii)$ The inverse image of comeager set is comeager.   I am studying Baire space and I would like to try the following facts  Can we collaborate to verify the above claims?    Thanks. Looking for a post-apocalyptic trilogy where pilot is nursed back to health by natives of the land I read a series of books around 2005. I think it might have been a trilogy.  Basic story line is in a post apocalyptic world. The main character is a pilot for the Federation or something similar to that. He crash lands, is nursed back to health by the natives, comes to like and respect them, and does not want to fight against them.  Also in the later book I think it also involves another race who are very Japanese-ish. They drink sake and the "clan" names are things like Fu-Jitsu and Mitsu-Bishi. How would I create an arch? Prove that the Interior of the Boundary is Empty Suppose X is a Metric Space  Let S $\subset X$  Prove that if S is Closed then, the Interior of the Boundary of S is Empty  Totally stuck on how to solve this. How do you overlay an image on a shader in Cycles? Why is it sensical for a proposition with a false antecedent to validate to true? In propositional logic, the statement "If pigs can fly, then elephants can lay eggs." validates to true because the antecedent validates to false.  In other words, given $a \rightarrow b$, if a is false, the entire statement is true. Why?  Just because the antecendent is false doesn't mean that another fact depends on it, right?