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idempotents acting as local identities Let $R$ be a ring with unity (not necessarily commutative) and $I$ an ideal of $R$.  Suppose that for every element $a \in I$ there exists an element $c\in I$ such that $ac=a$. Note that $c$ is related to $a$. Now we have the following question:  Can we say that  for every element  $a\in I$ there exists an idempotent element $c\in I$ such that $ac=a$?    Of course we have many examples such that the answer is true for them but in general  we don't know. I have a HP MFP127fw.  It's "supported" by Ubuntu 17.10 - except print jobs will not release from the print que because a special copywrited driver for HP is not installed.  I went through this with 16.10 and had to install a small HP driver that I found with Synaptic.  Then the jobs would release. It isn't part of HPLIPS.  Apparently Ubuntu 17,10 version doesn't run synaptic due to Wayland.  Where can I find this driver? Properties of p-adic valuations Let $n$ be a integer not equal to zero, $p$ a prime and $$|n|_p = p^{-\operatorname{ord}_p(n)}; |0|_p=0$$  Proposition: $|xy|_p=|x|_p|y|_p$  We want to show: $\operatorname{ord}_p(ab)=\operatorname{ord}_{p}(a)+\operatorname{ord}_p(b)$ So if we suppose there is a largest integral $v$ such that $p^{v}|(ab)$, then we can split up v such that $p^{v_1}|a,p^{v_2}|b$. If we suppose that there exist $v_1,v_2$ with them both being the largest integers such that $p^{v_1}|a, p^{v_2}|b$,  with $v_1+v_2= : v $ then $p^v|ab$. (I don't believe this to be a formal proof but rather my best own "idea" I can come up with...)  The script also mentions that the padic valuations' sums are bounded: $|x+y|_p\le \sup\{|x|_p,|y|_p\}$, how to show this?   Edit:   According to Robert Israel and Bill Dubuques answers, the first proposition is proven, if I understood correctly:   Given $ord(a)= v$ and $ord(b) = u$ which is the same as $a:= p^{v}m, b:= p^{u}n$ with $gcd(m,p)=gcd(n,p)=1$, because $ab=p^{u+v}mn$ with $gcd(mn,p)=1$ it follows that $ord(ab)=ord(a)+ord(b)$ and so $|xy|_{p}=p^{-ord(xy)}=p^{-(ord(x)+ord(y))}=|x|_{p}|y|_{p}$  Now to show: $|x+y|_{p}\le sup\{|x|_{p},|y|_{p} \}$  and the second can be tackled like this: set $a:=p^{v}m. b:=p^{u}n$, then checking two cases $1:(u=v)$ and $2:(v&lt;u)$: $(a+b)=p^{v}m+p^{u}n= p^{v}(m+p^{u-v}n)$ so in case of (u=v) it follows that  $ord(a+b)= u = sup\{ord(a),ord(b)\}$  and in case of $(v&lt;u)$ because the biggest v is $v:= u-1$: $ord(a+b)\le p^{u-1}(m+pn) \le p^{u}(m+n)= sup\{ord(a),ord(b)\}$  It seems that I did something wrong in both cases... Combo box in Java - nullpointer How should we handle downvote retaliation? How to get a list of all the labels in a LaTeX document in WinEdt I am writing a large document using LaTeX and WinEdt.  I have labeled each chapter, section and so on.  I would like to print out a list of just the labels (so that I don't duplicate one later on, and so that I have it handy in order to write \refs).    Is there a way to do this? How to NOT show keyboard layout chooser popup when changing language in GNOME 3 In a custom lightning component where one is using the recordEditForm tag, how can one validate fields that are filled by the user?  For example, in a quick action to create a Case record, we can have the following line inside the recordEditForm:  &lt;lightning:inputField aura:id="caseAccount" fieldName="AccountId"/&gt;   How can one enable or disable a button if certain fields are assigned by the user? Like if the account is specified, then a button on the component is enabled for the user.  Making the field required is not an option. There are many lengths in LaTeX. For instance, \enskip, \enspace, \quad, \parskip, \smallskip, ...  Some of them are mostly used for vertical spacing, some others are mostly used for horizontal spacing, and perhaps some of them are used in both cases. I wonder whether there is a list that indicates the meaning and intended use of each length in LaTeX.  By "intended use" I mean not only whether the length is vertical or horizontal, but in which circumstances it is a good practice to use that specific length. Add a time limit before you can vote on a post One image per overlay? Singular verb after two noun phrases joined by "and" that can be thought of as a single thing Basic mathematical induction regarding inequalities These are just the examples from my textbook, but I don't think it did not explain well.  One of the problem was to prove the inequality   $$n&lt;2^n$$  for all integers $n$.  I understand we assume if $P(n)$ is true, $P(k+1)$ is true.  Thus $k+1&lt; 2^{k+1}$  But in the text book, they added one more line that I don't understand:  $$k+1&lt;2^k+1 \le 2^k+2^k=2\cdot2^k=2^{k+1}$$  I understand  $$k+1&lt;2^{k+1}$$  But how do I know $2^k+1$ is larger than $k+1$, and smaller than $2^{k+1}$?  I have one another example just in case,  proving $2^n&lt;n!$ using induction.  $$2^{k+1}=2\cdot 2^k&lt;2\cdot k!&lt;(k+1)k!=(k+1)!$$  I understand we should start with $2^{k+1}&lt;(k+1)!$, but  Where did $2\cdot k!$ come from? Textbook says it came from inductive hypothesis, but I still don't get it.  Please advise. Direct way leads to nowhere.  $$\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} } = \lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{n \sqrt {1 +\frac{i}{n^2}} } = $$  $$ = \lim_{n\rightarrow \infty} \frac{1}{n} \cdot \lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {1 +\frac{i}{n^2}} } = 0 \cdot \sum^{n}_{i=1} 1 = 0 \cdot n $$  So, I tried to apply squeeze theorem:  $$\lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{\sqrt {n^2 +i} } \leq \lim_{n\rightarrow \infty} \sum^{n}_{i=1} \frac{1}{n}  = 1$$  Could you help me to figure out the lower bound? Short story collection about Transporter "Doors" I'm looking for a short story collection about different uses/inventions of transporter doors. I think it was just a single author collection, that I read in the early 2000s.  In each story these Transporter Doors exist, I seem to remember them being described as blank grey sheets. You step through them and come out somewhere else. The forward to the book talked about early Rocket stories where the launch of a spaceship would be the end of the tale.  Plots of the individual stories:  Two men one of whom's home is inside an asteroid are discussing the use of Base12 in their society and if it is because they have 12 fingers, one says that "people" used to use base10. The implication is that a different species arrived through the doors and replaced humans.  A man puts the code for his home into a door and is instead transported to prison where he's accused of murdering his wife, which he strongly denies.   Two scientists and a pilot take a probe ship deep into Jupiter's atmosphere, the will return to base via a door that they need to construct while they are down there. The scientists dislike each other and don't work well together. When they finish the door, the hamsters seem woosy out the other side, but the scientists insist it is ready. The pilot says he will go first as he can't fix it if it's wrong. When he arrives on the far side he's mentally damaged and the scientists have to get along without him acting as a buffer while the try to save themselves. Proving Pascal's Rule : ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$ I'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$.  I suppose I could use the counting rules in probability, perhaps combination= ${{n} \choose {r}}=\frac{n!}{r!(n-r!)}$.  I want to see an actual proof behind this equation.  Does anyone have any ideas? I'm having exactly the same issue as , also using Biber 2.14, albeit I didn't upgrade, downgrade, change TexLive version, change encoding or (seemingly) do anything. It's like Biber suddenly quit parsing dates on me.  Whether the bibfile is encoded in ASCII or UTF-8 makes no difference. Whether the date is a short format or the full YYYY-MM-DD makes no difference. I'm not sure what else to test.  Here's a MWE and the associated .blg file:  \begin{filecontents}{mwe.bib}     @Article{JD2020,         author       = {John Doe},         title        = {A Title},         date         = {2020-01-01},         %date       = {2020}         journaltitle = {Journal},     } \end{filecontents}  \documentclass{article} \usepackage{biblatex} \addbibresource{mwe.bib}  \begin{document}     test\autocite{JD2020}     \printbibliography \end{document}   [1] Config.pm:304&gt; INFO - This is Biber 2.14 [1] Config.pm:307&gt; INFO - Logfile is 'mwe.blg' [154] biber-darwin:322&gt; INFO - === Thu Apr 30, 2020, 11:04:33 [185] Biber.pm:415&gt; INFO - Reading 'mwe.bcf' [354] Biber.pm:943&gt; INFO - Found 1 citekeys in bib section 0 [384] Biber.pm:4256&gt; INFO - Processing section 0 [385] Utils.pm:75&gt; INFO - Globbing data source 'mwe.bib' [385] Utils.pm:91&gt; INFO - Globbed data source 'mwe.bib' to mwe.bib [401] Biber.pm:4455&gt; INFO - Looking for bibtex format file 'mwe.bib' for section 0 [402] bibtex.pm:1653&gt; INFO - LaTeX decoding ... [406] bibtex.pm:1471&gt; INFO - Found BibTeX data source 'mwe.bib' [437] Utils.pm:304&gt; WARN - Entry 'JD2020' (mwe.bib): Invalid format '2020-01-01' of date field 'date' - ignoring [457] UCollate.pm:68&gt; INFO - Overriding locale 'en-US' defaults 'normalization = NFD' with 'normalization = prenormalized' [457] UCollate.pm:68&gt; INFO - Overriding locale 'en-US' defaults 'variable = shifted' with 'variable = non-ignorable' [457] Biber.pm:4084&gt; INFO - Sorting list 'nty/global//global/global' of type 'entry' with template 'nty' and locale 'en-US' [457] Biber.pm:4090&gt; INFO - No sort tailoring available for locale 'en-US' [461] bbl.pm:652&gt; INFO - Writing 'mwe.bbl' with encoding 'UTF-8' [463] bbl.pm:755&gt; INFO - Output to mwe.bbl [463] Biber.pm:128&gt; INFO - WARNINGS: 1   The \printbibliography output actually looks like this: [1] John Doe. “A Title”. In: Journal ().  In my actual thesis, all the dates are gone from the pdf output. Why is 'ping' unable to resolve a name when 'nslookup' works fine? On my Windows XP workstation, I can find the machine I want to connect to in DNS with nslookup:  nslookup wolfman Server: dns.company.com Address: 192.168.1.38  Name: wolfman.company.com Address: 192.168.1.178   But, when I try to connect to that machine, I get an error telling me that the machine can't be found (i.e., can't be looked up in DNS):  C:\&gt; ping wolfman Ping request could not find host wolfman. Please check the name and try again.   I am able to connect if I use the IP address directly:  C:\&gt; ping 192.168.1.178  Pinging 192.168.1.178 with 32 bytes of data:  Reply from 192.168.1.178: bytes=32 time=41ms TTL=126 Reply from 192.168.1.178: bytes=32 time=41ms TTL=126 Reply from 192.168.1.178: bytes=32 time=44ms TTL=126 Reply from 192.168.1.178: bytes=32 time=38ms TTL=126   I could work around this by adding an entry to my hosts file, but I would rather find out why this is happening.  The problem is transient, most of the day I can connect to the machine just fine.  How is this possible?  ETA: I left this out for brevity, but it was asked for:  C:\&gt; ping wolfman.company.com Ping request could not find host wolfman.company.com. Please check the name and try again.   ETA: Other applications get the same results.  I only tried ping to simplify.  telnet can't connect, Cygwin apps print a "unknown host wolfman" message.  Update: Using wireshark, I found that my workstation is not attempting a DNS lookup.  It's just reporting the "could not find host" error message. Accidentally turned on talkback I accidentally turned on talkback in the Accessibility menu. When I turned it on I got some kind of walk through on how to use it. I couldn't cancel it. So I turned off my phone.   When I turned it back on again it took me 15 minutes to unlock my phone. Apparently I had to touch a button to give it focus and then double tap it to click it.   Now my question is: how do I turn this off? I can't reach the settings because I have don't know how to access a drop down and I can't scroll through all my apps. Why don't metals bond when touched together? It is my understanding that metals are a crystal lattice of ions, held together by delocalized electrons, which move freely through the lattice (and conduct electricity, heat, etc.).   If two pieces of the same metal are touched together, why don't they bond?   It seems to me the delocalized electrons would move from one metal to the other, and extend the bond, holding the two pieces together. If the electrons don't move freely from one piece to the other, why would this not happen when a current is applied (through the two pieces)?