Let $R$ be a ring with identity. Let $u \in R$ has at least one right inverse. Show that if $u$ has exactly one right inverse, then $u$ is a unit.
Suppose $u$ is an element of a ring with a right inverse. I'm trying to understand why the following are equivalent.   $u$ has at least two right inverses $u$ is a left zero divisor $u$ is not a unit   If $v$ and $w$ are distinct right inverse of $u$, then $u(v-w)=0$, but $v-w\neq 0$, so $u$ is a left zero divisor. It's also clear that if $u$ is a left zero divisor, it cannot be a unit (else I could cancel $u$ from $ub=0$ to see $b=0$).  I'm having a heck of a time seeing why $u$ is not a unit implies $u$ has at least two right inverses. I tried the contrapositive, but saw no good approach. What am I missing?
Can "Wow!" be a sentence? The question is clear. Can "Wow!" be a sentence?  Imagine a hypothetical context where I'd utter something like this:     Wow! What an amazing idea! I should think about a way to push this fix.   We see "wow" in casual speech often. It bears punctuation marks (typically an exclamation mark, denoting the rising intonation) and a capital letter &mdash; just like all "regular" sentences.  On the other hand, neither Wow! nor What an amazing idea! contain any verbs. These certainly don't have any parts of speech in the other sentence, so what are they? Are they sentences or clauses? In other words, can "Wow!" be a sentence despite its lack of [any implied] verbs, and subsequently lack of any distinguishable predicates?  Please back up your answers with references. How to factory reset without data loss and undue pain? How do I resolve these unmet dependencies? 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Not sure whether is a new trend, or something to do with my browser settings (I haven't changed them lately, AFAIK), or Google Chrome, which is my browser. This is on Windows 7 running on a Dell Vostro laptop. I added the info about the laptop because I suspect the problem may even be due to my hardware settings, since I found the default brightness to be too high after I bought it, and changed it via Control Panel.  Would like to know what are some of the things I can check or change to solve this.  Thanks. Closed unit interval is connected proof I've been asked the following question:  What five odd integers from the set  $\{1, 3, 5, 7, 9, 11, 13, 15\}$ that when summed together equals to $30$? Note that any integer can be used more than once.   If my limited knowledge of maths is correct, there should be no answer, as no odd number of odd integers summed together can give an even number. 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Here is an  on Sports:    These screenshots were all taken on Firefox, but I am seeing the same problems in Chrome and Internet Explorer.  One more example: How to use manual partitioning during installation? Finding the 1st three derivatives of a function and plotting all four curves