Let $A$ be a subring of a ring $B$. Suppose $B$ is integral over $A$. Let $\Omega$ be an algebraically closed field. Then every homomorphism $\psi\colon A \rightarrow \Omega$ can be extended to a homomorphism  $\phi\colon B \rightarrow \Omega$?
Let $A \subset B$ be a finite ring extension. Fix a ring homomorphism $\nu : A \to \Omega$, where $\Omega$ is an algebraically closed field.   I want to show there exists a non-zero homomorphism $v : B \to \Omega$ of $A$-modules.  I want show that the space $V$ of all the homomorphisms $v$ in (i) is a non-zero finite dimensional vector space over $\Omega$. Show that the algebra $B$ acts on this space. Show that eigenvectors of this action correspond to homomorphisms of algebras $\lambda : B \to \Omega$ that extend the homomorphism $\nu$. Deduce that the set of such homomorphisms $\lambda : B \to \Omega$ is finite and non-empty.    We can assume that $B = Ab_1 + ... + Ab_n$ where $b_1,...,b_n \in B$. The chose $v(b_i)=\omega_i \in \Omega-\{0\}$ and then define $$ v( a_1 b_1 + ... + a_n b_n ) = \nu(a_1) v(b_1) + ... + \nu(a_n) v_n = \nu(a_1) \omega_1 + ... + \nu(a_n) \omega_n \ .$$ This is an $A$-module homomorphism. Am I correct? For the second part, that $V$ is not-zero follows from the first part. That it is finite dimensional follows from the fact that $v_i(b_j)=\delta_{i,j}$ (Kronecker's delta) with $i=1,...,n$ spans $V$ and hence $\dim V \le n &lt; \infty$. I define the action of $B$ over $V$ to be $$b \bullet v = \left( x \mapsto v(b)v(x) \right) \mbox{  for all } b,x \in B \ .$$ This seems to be the natural action to define. Am I correct? However, I haven't managed to solved the rest (the eigenvector stuff and the relation to $\nu$).
Set limit to one author when using "et al." in biblatex In the header of my document I set the maximal author names to 2:  \usepackage[style=authoryear-icomp, maxbibnames=9, maxcitenames=2, backend=biber]{biblatex}   Now, when I cite texts with two authors, LaTeX gives me the following:     (Author_A/ Author_B 2012: 232)   When I have three authors or more, LaTeX makes it:     (Author_A/ Author_B et al. 2012: 232)   But I want LaTeX to just name the first author if there are more than two authors ... like this:     (Author_A et al. 2012: 232)   Any tips on how to achieve this ? Non-trivial cyclic group with unique generator Let $G$ be a non-trivial cyclic group with an unique generator. Show that $G$ has exactly two elements.   Let $g$ be a generator of $G$. My first idea to approach the problem was to show that if $|G|=n &gt; 2$ then there is an integer $k$ between 1 and $n$ such that $\langle g^k \rangle = G$ or, equivalently, $g \in \langle g^k \rangle$. My second idea was to analyse the problem in 2 separate cases. For the first one, if $n$ is prime, let $k$ be a positive integer then, by Lagrange's Theorem, $|g^k| \mid n$ and, therefore, $\langle g^k \rangle = n$. Hence, we must conclude that, if $n &gt; 2$ the generator cannot be unique. For the second case, where $n$ is composite, I was hoping to get to a contradiction, but could not advance any further. Is there a way to develop either of these ideas or is it necessary a new approach? I'm a mathematician interested in abstract QFT.  I'm trying to undersand why, under certain (all?) circumstances, we must have $T^2 = -1$ rather than $T^2 = +1$, where $T$ is the time reversal operator.  I understand from the Wikipedia article that requiring that energy stay positive forces $T$ to be represented by an anti-unitary operator.  But I don't see how this forces $T^2=-1$.  (Or maybe it doesn't force it, it merely allows it?)  Here's another version of my question.  There are two distinct double covers of the Lie group $O(n)$ which restrict to the familiar $Spin(n)\to SO(n)$ cover on $SO(n)$; they are called $Pin_+(n)$ and $Pin_-(n)$.  If $R\in O(n)$ is a reflection and $\tilde{R}\in Pin_\pm(n)$ covers $R$, then $\tilde{R}^2 = \pm 1$.  So saying that $T^2=-1$ means we are in $Pin_-$ rather than $Pin_+$.  (I'm assuming Euclidean signature here.)  My question (version 2): Under what circumstances are we forced to use $Pin_-$ rather than $Pin_+$ here?  (I posted a  on physics.stackexchange.com last week, but there were no replies.)    EDIT: Thanks to the half-integer spin hint in the comments below, I was able to do a more effective web search.  If I understand correctly, Kramer's theorem says that for even-dimensional (half integer spin) representations of the Spin group, $T$ must satisfy $T^2=-1$, while for the odd-dimensional representations (integer spin), we have $T^2=1$.  I guess at this point it becomes a straightforward question in representation theory: Given an irreducible representation of $Spin(n)$, we can ask whether it is possible to extend it to $Pin_-(n)$ (or $Pin_+(n)$) so that the lifted reflections $\tilde R$ (e.g. $T$) act as an anti-unitary operator. It appears that in anticipation of D&amp;D Next Wizards of the Coast has removed much of the Dungeons and Dragons, 3rd Edition Web content. I didn't perform a thorough search to determine if all the content is removed (e.g. I know the Three Dragon Ante Web enhancement  and the Mind's Eye column about psionic tattoos  are both gone--better not to ask why those are my go-to articles), but it all might be.  What's the best way--and by best I mean both legal and convenient--to continue to access this content? How do I install Ubuntu to a USB key? (without using Startup Disk Creator) Indenting a whole paragraph How do I indent a whole paragraph at the same distance from my left margin, with the simplest command or environment? Two "offtopic" flag reasons Question:  Is exception handling in Java actually slow?  Conventional wisdom, as well as a lot of Google results, says that exceptional logic shouldn't be used for normal program flow in Java.  Two reasons are usually given,   it is really slow - even an order of magnitude slower than regular code (the reasons given vary),     and    it is messy because people expect only errors to be handled in exceptional code.     This question is about #1.  As an example,  describes Java exception handling as "very slow" and relates the slowness to the creation of the exception message string - "this string is then used in creating the exception object that is thrown. This is not fast."  The article  says that "the reason for this is due to the object creation aspect of exception handling, which thereby makes throwing exceptions inherently slow".  Another reason out there is that the stack trace generation is what slows it down.  My testing (using Java 1.6.0_07, Java HotSpot 10.0, on 32 bit Linux), indicates that exception handling is no slower than regular code.  I tried running a method in a loop that executes some code.  At the end of the method, I use a boolean to indicate whether to return or throw.  This way the actual processing is the same. I tried running the methods in different orders and averaging my test times, thinking it may have been the JVM warming up.  In all my tests, the throw was at least as fast as the return, if not faster (up to 3.1% faster).  I am completely open to the possibility that my tests were wrong, but I haven't seen anything out there in the way of the code sample, test comparisons, or results in the last year or two that show exception handling in Java to actually be slow.  What leads me down this path was an API I needed to use that threw exceptions as part of normal control logic.  I wanted to correct them in their usage, but now I may not be able to.  Will I instead have to praise them on their forward thinking?  In the paper , the authors suggest that the presence of exception handlers alone, even if no exceptions are thrown, is enough to prevent the JIT compiler from optimizing the code properly, thus slowing it down.  I haven't tested this theory yet. How to prove that for all natural numbers, $4^n > n^3$? This is a problem set I have, it's not a homework but it's very important practice...  Send me some hints please, I don't want an answer I need to get it by myself but I'm failing miserably...   The problem is:     Prove that for all natural numbers $n$,  $4^n &gt; n^3$ How to Set a Short User Password in macOS Mojave and Later (10.14+) I just installed macOS Mojave and put in my usual short desktop-only password I've been using for years (yes, you read that right). Now though, the operating system rejects setting a password less than 4 characters. It also does not allow having no password at all.  I am completely comfortable using the shell if need be.  How can I change the password policy to allow quite short passwords? Show that quotient rings are not isomorphic I've been given a homework problem that requires me to show that the rings $\mathbb{C}[x,y]/(y - x^2)$ and $\mathbb{C}[x,y]/(xy-1)$ are not isomorphic.    This is my attempt at a solution:  For $\mathbb{C}[x,y]/(y - x^2)$, we can parametrize in the following way: $x = t$ and $y = t^2$. Then this ring is isomorphic to $\mathbb{C}[t]$.  For $\mathbb{C}[x,y]/(xy-1)$, we can parametrize $x = t$ and $ y = 1/t$. Then this is isomorphic to $\mathbb{C}[t, 1/t]$.   But $\mathbb{C}[t, 1/t]$ is not isomorphic to $\mathbb{C}[t]$.  Am I on the right track? If not, any helpful hints? Difference between Groovy Binary and Source release? "Use Option as Meta key" with ability to type the # symbol I am using Terminal version 2.5, where I have two requirements:  I need to be able to type the Meta key. I need to be able to type the # character, too (Option+3 for the UK keyboard layout)  How can I satisfy both of these? By default, (2) is satisfied, but (1) is not. I can then turn on &quot;Use Option as Meta key&quot;, which satisfies (1), but breaks requirement (2). How can I have both? (It shouldn't be so hard for me to just be able to type.) Edit: what I'm looking for, for example, might be a way to just &quot;Use Option as Meta key&quot; for the left/right Option key, or a way to override the &quot;Use Option as Meta key&quot; for the sole exception of the # character. Prove that the range of the entire function $z^2+\cos(z)$ is all of $\mathbb{C}$. Prove that the range of the entire function $z^2+\cos(z)$ is all of $\mathbb{C}$.   I'm aware this question has been  already, but the explanations were a little shakey and referenced a google sample preview of a text of whose terminology I wasn't very familiar with, so I would like some more input if possible.  Here's my attempt:  Let $f(z)=z^2+\cos(z)$. Through the power series of the cosine function $$ f(z)=z^2+1-\frac{z^2}{2!}+\frac{z^4}{4!}-\cdots=1+\frac{z^2}{2!}+\frac{z^4}{4!}-\cdots, $$ so $$ g(z):=\frac{f(z)-1}{z} $$ has a removable singularity at $z=0$ and is effectively an entire odd function. Picard's little theorem asserts that the range of $g$ is all of $\mathbb{C} $: if there exists some $z_0$ such that $g(z)\neq z_0$ for all $z\in\mathbb{C}$, then $-z_0$ is also missed by oddness of $g$, which is a contradiction (since $g(0)=0$ we rule the possibility that $z_0=0$). (I'm not sure about the following step) It follows that $zg(z)$ and so $zg(z)+1=f(z)$ is surjective.  Is this method valid? Would I need further justification to show that $z\mapsto z$ and $z\mapsto g(z)$ surjective imply its product -- $z\mapsto zg(z)$ -- is also surjective (if this is even true). Any input is greatly appreciated.  Edit: After thinking about this for a moment longer, I realize this logic would also show that the range of any entire even function is all of $\mathbb{C}$, which seems like quite a strong conclusion. This makes feel like there are holes in my logic. I have Opportunity-lookup field on Opportunity (self),  Scenario:   Opportunity A if select Opportunity B using lookup then Opportunity B's same lookup field automatically populate with Opportunity A.   I tried to achieve this using trigger - before insert/update but its failing as:  System.DmlException: Update failed. First exception on row 0 with id 006G000000DHaBlIAL; first error: SELF_REFERENCE_FROM_TRIGGER, Object (id = 006G000000DHaBl) is currently in trigger Oppy_before_INSUPD, therefore it cannot recursively update itself: [] 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? When drag force ($bV$) equals to object's weight (mg) then upward and downward force becomes equal. As a result the object comes to rest. If this is true, how is a body moving with constant velocity? How does JavaScript .prototype work? Just read this which basically says you can freeze and thaw food in the fridge any number of times.    I'm still unclear on one point. If I had food which would be good in the fridge for 4 days and i put it in the freezer after 3 days, when I put it back in the fridge is my food good after its thawed for another 4 days or 1 more day? Evaluate the limit: $$\mathop {\lim }\limits_{x \to \infty } x\left( {{{\left( {1 + {1 \over x}} \right)}^x} - e} \right)$$  My attempts didn't yield a result. I'd be glad for a guidance. Thanks!