Let's say we have a commutative ring $R$ with a unit and elements $a,b \in R$ which divide each other, e.g. $a | b$ and $b|a$. The question is now:     Does there exist an invertible element $\epsilon \in R^\times$ with $a \epsilon = b$ or are there counterexamples e.g. rings and elements as above, for which you can't find such an $\epsilon$?   This question arose from an exercise I gave to my students for commutative rings without unit (where there are plenty counterexamples, e.g. $R=\mathbb Z \times 2\mathbb Z$). Since we couldn't find counterexamples or a proof for unital rings I am asking here the question.  Stuff we already tried/figured out:   It is quite clear, that in a counterexample $a$ and $b$ are zerodivisors. It is useless to consider cross-products of rings, since you can solve the problem pointwise. $\mathbb Z / n \mathbb Z$ are no counterexamples.
Recall the following relevant definitions. We say that   $b$ is divisible by $a$ in $R$, or $a\mid b$ in $R$, if $b = r a$ for some $r\in R$. $a$ and $b$ are associates  in $R$ if $a\mid b$ and $b\mid a$, (or, equivalently, if $aR = bR$). $u\in R$ is a unit if it has a multiplicative inverse (a $v\in R$ such that $uv=vu=1$). $a$ and $b$ are unit multiples in $R$  if $a = ub$ for some unit $u\in R$.   Given these definitions, my question is,     If $R$ is a commutative ring with unity and $a,b\in R$ are associates in $R$, are $a$ and $b$ unit multiples in $R$?   I was told that this not always true. But I encountered some difficulties in finding a counterexample.
Which package to use for presentations? Beamer, Prosper, or Other scope resolution operator without a scope I have a program which spawns several child processes.  I want to know which files this program and its children are creating, deleting, and modifying.   inotifywait can tell me which files are being accessed, but cannot tell me which process is doing it. And I have to know where the files are being created in order to set up the monitor. auditctl will let me log which processes modify a file, but, again, I have to know where the files are in order to set up logging. lsof will tell me all of the files, but doesn't seem to be a good solution for real-time tracking.   Is there anything that will let me monitor a program's file manipulation without knowing a priori which files will be manipulated? Is there a word to describe words that can be spelled different ways? Is there a word in English to describe words that can be spelled more than one way?  For example:   colour and color favourite and favorite   However, this shouldn't be limited to words spelled differently in different dialects, though I'm not sure what other case there would be. When installing Wine on 64-bit Ubuntu 12.04, both 64-bit support and 32-bit support get installed. If I run wine foo.exe where foo.exe is a 64-bit-aware installer, it thinks it's on 64-bit Windows. This would be fine if this mode didn't fail, but in my case, it does (yet the Wine database suggests the app I'm trying to install should work, presumably as a 32-bit app).  How do I tell Wine (and Winetricks) to use Wine in 32-bit-only mode? How do I get HTML5 h.264 Video working on Firefox? According the the , h.264 video should now be working in Linux.  It does also say the appropriate gstreamer plug-ins need to be installed.  Does anyone know what these are?    I used this  and the  to verify that h.264 support works in Google Chrome on my Ubuntu 12.04 64-bit, but does not work on Firefox 26. redraw GUI on didreceivelocalnotification when come from lock screen iphone Cleanest way to toggle a boolean variable in Java? Is there a better way to negate a boolean in Java than a simple if-else?  if (theBoolean) {     theBoolean = false; } else {     theBoolean = true; } Word for a male that prefers female companions (no slang such as "pimp") Why is uniq ignoring Unicode and lines with a single letter? I'm trying to combine both the American and British dictionaries into one large dictionaries, and I'm trying to remove all the duplicates from the superset, but it seems that uniq is not outputting words like, "épée" or single letters.  This is what I've tried using:  LC_COLLATE=en_US.UTF-8 cat american-english british-english |sort|uniq -u &gt; unique_sorted_combined_dict   If I just do this:  LC_COLLATE=en_US.UTF-8 cat american-english british-english |sort &gt; sorted_combined_dict   "épée" and other such words do show up, along with single letters.  Is there something I'm missing here with uniq?  I should note that I'm using uniq from the GNU coreutils on Ubuntu 12.10, if that makes any difference. Can you break up your Attack action for a bonus action? You can do it to move:     If you take an action that includes more than one weapon attack, you can break up your movement even further by moving between those attacks. For example, a fighter who can make two attacks with the Extra Attack feature and who has a speed of 25 feet could move 10 feet, make an attack, move 15 feet, and then attack again. (PHB 190)   It would be necessary for the Great Weapon Master feat:     On your turn, when you score a critical hit with a melee weapon or reduce a creature to 0 hit points with one, you can make one melee weapon attack as a bonus action   As a level 17 Fighter, if I can not place a bonus action between attacks, there are two possibilities:   I have to wait until I finished all 4 attacks of the action, to make another attack as a bonus action the bonus action attack is wasted unless the kill or crit happens on the 4th attack   So can I take a bonus action in the middle of the attacks of an action? Is the tag info page not available on mobile theme? I was surfing SO on my iPhone, and wanted to see the info (tag-wiki) page for a tag, but didn't find it. Clicking on the tag always leads to the list of question, and there is no other link to the tag wiki. Is it not available on the mobile site? Joint density for exponential distribution Let $X_1$ and $X_2$ be independent random variables each having a exponential distribution with mean $\lambda = 1$.  (a) Find the joint density of $Y_1 = X_1$ and $Y_2 = X_1 + X_2$.  (b) Get the marginal density of $f_1(y_1)$ and $f_2(y_2)$.  $f(x_1)=e^{-x_1}$, $f(x_2)=e^{-x_2}$  Since $X_1$ and $X_2$ are independent, $f(x_1, x_2)=f(x_1)f(x_2)=e^{-x_1-x_2}$  However, how could I find the joint density?  I mean, it is quite easy to find marginal ones from the joint one, but how can I do the opposite? How prove this nice limit $\lim\limits_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$ Nice problem:  Let $a_{0}=1$ and $$a_{n}=a_{\left\lfloor n/2\right\rfloor}+a_{\left\lfloor n/3 \right\rfloor}+a_{\left\lfloor n/6\right\rfloor}.$$  Show that  $$\lim_{n\to\infty}\dfrac{a_{n}}{n}=\dfrac{12}{\log{432}},$$  where $\lfloor x \rfloor$ is the largest integer not greater than $x$.  It is said this problem was created by Paul Erdős, and I can't find this problem's solution, does anyone have any nice methods? Thank you. How can I tell what year my bike was made? I would like to set a maximum execution time for sql queries like set_time_limit() in php. How can I do ? I'm fairly excited for Unity, as it looks like a promising new direction for Ubuntu.  However, I do have a concern - will it be possible to use Unity without the global menu?  I have my window manager set to focus-follows-mouse/sloppy focus, and find the productivity gains to be immense. Sloppy focus is incompatible, however, with global menus, as it is possible for the focus to change while you move from window to menu.  Will Unity support an option to use window menus while still using Unity? How many and which are the uses of "const" in C++? As a novice C++ programmer there are some constructs that look still very obscure to me, one of these is const. You can use it in so many places and with so many different effects that is nearly impossible for a beginner to come out alive. Will some C++ guru explain once forever the various uses and whether and/or why not to use them? Not only in ortho: top, front, back.  Please don't link this answer   I want a background image like this guy has -> Handling registration ID changes in Google Cloud Messaging on Android