A surjective ring homomorphism $\phi : C([0,1]) \rightarrow \mathbb{R}$ is evaluation at a point
Characterizing all ring homomorphisms $C[0,1]\to\mathbb{R}$.
What are the differences between a multidimensional array and an array of arrays in C#? What are the differences between multidimensional arrays double[,] and array-of-arrays double[][] in C#?  If there is a difference, what is the best use for each one? Show that $a \star b=a \cdot b+a+b$ is binary operation for the group $\Bbb{ Q} \setminus \{-1\}$ The group $\left(\Bbb{ Q}\setminus\{-1\},\star\right)$ has as its underlying set the rational numbers different from $-1$ and the operation $\star$ is defined as $a \star b=a \cdot b+a+b$ where multiplication/addition are the usual operations with rational numbers.Show that this is a binary operation.  What I did (Please help verify):  identity element has to be 0 since $a\star 0 = a+0+a(0)$ so for $a\star b=0$ we have  $ab+a+b=0$ $$ab+a=-b \\ a(b+1)=-b \\ a=\frac{-b}{b+1}$$ so $b$ cant be $-1$ since that solution is not in $\Bbb{Q}$. Similar question might have been answered here. Is this operation true for all groups? I think so.  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Related questions that don't quite cover my question: On Takodana,      the main characters and various bar patrons can see the planet Hosnian Prime being destroyed by Starkiller Base with their (respective) naked eyes. How did they see this? Is the Hosnian system close enough to see? If it's not very close and they could just see the explosion, how did the light from it reach them so quickly? How can I control the name of generic WCF return types? I've got a WCF Web Service method whose prototype is:  [OperationContract] Response&lt;List&lt;Customer&gt;&gt; GetCustomers();   When I add the service reference to a client, Visual Studio (2005) creates a type called "ResponseOfArrayOfCustomerrleXg3IC" that is a wrapper for "Response&lt;List&lt;Customer>>".  Is there any way I can control the wrapper name?  ResponseOfArrayOfCustomerrleXg3IC doesn't sound very appealing... How to resolve BackgroundContainer.dll error on startup? When I start my desktop it gives me an error about BackgroundContainer.dll.    How to get rid of it?  I found nothing when i started regedit.exe and searched (F3 or CTRL+F) for BackgroundContainer.dll Since version 3.2, uTorrent displays ADs in the form of a "Featured Torrent" banner and a small square advertising in the sidebar. Is there a way to disable those? I've been using Texmaker for ~5 years now and prefer it over other editors because of its flexibility.   Is there a way to auto-indent LaTeX code in TeXmaker so that the code becomes more manageable ? How do I boot into recovery mode? Should one downvote answers to off-topic questions? It annoys me when people post answers to obviously off-topic questions like , especially after those questions have already received several close votes. Migrations seem to happen rather infrequently these days, so there's probably no reputation to gain for questions that are likely to be closed. And although sure, the answer may help the OP, it also encourages bad behavior (posting off-topic questions.)  Is it appropriate to downvote these answers?   Personally, I think I could make a case for both sides. Why is it so difficult to pull an immersed mug out of the water? Suppose I put a mug in the water and then tilt it so that the water gets inside. It then floats due to buoyant force. But if I try to pull it out then a lot of force is needed. That's due to suction? What exactly happens at the moment when the force I apply is enough to pull the mug out? And what causes the loud pop sound when that happens? Why do we not use the definite article in "Where can I find the room 401?" Last week in college, a student [with a thick South Asian accent] asked me:     "Excuse me, where can I find the room 401?".   I realized that to my native English ears, the word "the" sounded non-native. This had me realize that we sometimes use no article.  I am learning French; in French, we seem to always use some kind of article before nouns. It surprised me, at that moment, to realize that English doesn't always do the same! (That, and using the definite article in "the room 401" seems to make sense if I think about when definite articles are used).  Is there a grammatical name or concept for this "no article" aspect of our language? Are there any websites that can explain this to me?