Anyone knowns an example of a R-module finite genereted with a submodule not finite generated?  I find the following example: Taking the set of function $f:[0,1]\rightarrow\mathbb{R}$ seen as module of it self. This is finite generated. If we take the subset of functions $f$ such that $f(x)=0$  for all $x\in[0,1]$ except a finite numbers of points then we will get a submodule wich is not finite generated. Why?  This is a different example of the others I saw.   Thanks a lot!!!
I'm looking for an example of a finitely generated module with submodules that are not finitely generated.  I've found  dealing with group (i.e. an example of a finitely generated group with subgroups that are not finitely generated). But I can't figure out whether that question do help to this one.  And I actually want to find a more "module-like" example rather than an example derived from a 'strange' group.  Can you please help? Thank you!
Reputation does not update after community wiki is removed from a post, and does not match the value displayed in the reputation audit If votes were cast on a post while it was community wiki, and a moderator removes wiki from that post, the votes should count towards reputation again. However, while this is visible in a reputation audit, , and the reputation score needs to be recalculated to reflect the change.  Can someone fix this? Steam not opening in Ubuntu 16.04 LTS Proof on Catalan numbers Why does Oracle 9i treat an empty string as NULL? I know that it does consider ' ' as NULL, but that doesn't do much to tell me why this is the case.  As I understand the SQL specifications, ' ' is not the same as NULL -- one is a valid datum, and the other is indicating the absence of that same information.  Feel free to speculate, but please indicate if that's the case.  If there's anyone from Oracle who can comment on it, that'd be fantastic! New top bar quick reference guide / Where has the link to  gone? Since the new top bar has been rolled out - a number of different people, established and new users alike, have asked questions on where a specific link has disappeared to with the new top bar design. Where has the link to &lt;X&gt; gone? Linux - why do the su and sudo binaries need to be set-UID root? I am trying to find the correct reason for the stated question. My understanding is that:   sudo needs to read the /etc/sudoers file which is only readable by root, which is why it needs to be set-UID root su is going to create a new shell with a different real and effective UID, and needs to check the password. To check the password, it needs to read /etc/shadow, which is why it needs to be set-UID root. After checking password, it would need to call setuid() on the forked process, and to use an arbitrary UID argument, its parent process must have root as effective UID, so this also makes another reason.   Are the above reasons correct ? I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the proper orthochronous Lorentz group $SO(1,3)_+^{\uparrow}$ is given by \begin{equation} [J_i,J_j]=i \epsilon_{ijk} J_k \end{equation} \begin{equation} [J_i,K_j]=i \epsilon_{ijk} K_k \end{equation} \begin{equation} [K_i,K_j]=- i \epsilon_{ijk} J_k \end{equation} We can now define new generators with the old ones $N^{\pm}_i= \frac{1}{2}(J_i \pm i K_i)$ which satisfy \begin{equation} [N^{+}_i,N^{+}_j] = i \epsilon_{ijk} N^{+}_k ,\end{equation} \begin{equation} [N^{-}_i,N^{-}_j] = i \epsilon_{ijk} N^{-}_k ,\end{equation} \begin{equation} [N^{+}_i,N^{-}_j] = 0. \end{equation} where we can see that $N^{+}_i$ and $N^{-}_i$ make up a copy of the Lie algebra $\mathfrak{su}(2)$ each. My problem is to get what is going one here mathematically precise. Are the following statements correct and if not why:  When we build the new operators from the old generators we complexified $\mathfrak{so}(1,3)_+^{\uparrow}$ \begin{equation}(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} = \mathfrak{so}(1,3)_+^{\uparrow}\otimes \mathbb{C} \end{equation} We saw that $\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C}$ is isomorph to two copies of the complexified Lie algebra of $\mathfrak{su(2)}$: $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq \mathfrak{su(2)}_{\mathbb{C}} \oplus \mathfrak{su(2)}_{\mathbb{C}} $. Where exactly did we need that $\mathfrak{su(2)}$ is complexified here? The Lie algebras defined by $N^{\pm}_i$ are exactly those of $\mathfrak{su(2)}$ and we never use complex linear combination of $N^{\pm}_i$ or am I wrong here? $\mathfrak{su(2)}_{\mathbb{C}}$ is isomorph to $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$: \begin{equation}\mathfrak{su(2)}_{\mathbb{C}} \simeq (\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C} \end{equation} Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ denotes the complexified Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ correct?  Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ denotes the real Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C} \oplus (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ correct?  I looked this topic up in different books and each seemed to state something different. One book even used three differrent versions of $\mathfrak{sl}(2,\mathbb{C}) $ namely: $\mathfrak{sl}(2,\mathbb{C}) $, $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$ and $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{R}$.  states simply that $\mathfrak{sl}(2,\mathbb{C}) $ is the complexification of $\mathfrak{su(2)}$ without making any reference to $SL(2,\mathbb{C})$ which does not help me either. Any help would be great. Almost sure convergence of the Poisson process Let $N = \{N(t) \}_{t\geq 0 }$ be a Poisson process. I already know that $N(t)- \lambda t$ is a martingale where $\mathbb{E} [ N(t) ] = \lambda t$.   I want to prove that $$ \frac{N(t)}{t} \rightarrow \lambda, \quad t \rightarrow \infty \quad \text{ a.s.}. $$  To prove this I received a hint that I should use the martingale law of large numbers which states that:  Theorem: (Martingale LLN)  Let $S= (S_n = \sum_{k=1}^n X_k)_{n=1,2,\ldots}$ be a martingale with respect to the filtration $(\mathcal{F}_n)_{n=1,2,\ldots}$. Then $\sum_{k=1}^n X_k/k$ converges almost surely on the set $\left\{ \sum_{k=1}^\infty k^{-2} \mathbb{E}\left[ X_k^2 \mid \mathcal{F}_{k-1} \right] &lt; \infty \right\}$. Hence $S_n/n \rightarrow 0$ a.s. on the same set.  Now to use this theorem I think I need to use Doob-Meyer to decompose $N(t)$ as the sum of a martingale and an increasing process. Which would be the martingale $M(t) = N(t) - \lambda t$ and increasing process $A(t) = \lambda t$. But how is $M(t)$ the sum of a martingale? And I do not know how this exactly relates to the thing I have to prove. Thanks for any help. Sign of Work and potential energy in electrostatics Conceptual question:  Suppose we have a configuration of point charges. If the potential of the energy of the system is negative, this means work is positive.   I'm kind of rusty with my mechanics, does the fact that the work is positive mean that to arrange the configuration you have to do work against the electric field? I.e. supply energy via some external force?  In terms of electrostatics, is there a general way of interpreting the sign of potential energy and work? Is it possible to set DNS for mobile network Why high voltage transmission lines? I'm currently trying to configure our exchange 2010 to be used as a external smtp relay. What I'm trying to do is allow certain websites that we manage to send email using our exchange server smtp relay. I was told that I needed to created a receiver connector under Server Configuration>Hub Transport. After I created the receiver connector I added the following confi in which I added the website IP address in the "Receiver mail from remote servers ect" then unders Authentication I selected Externally Secured,TLS and under permissions I selected exchange servers.   I went to one of our websites and configure it with this configuration and I'm still getting "SMTP connect failed."  Can someone help me? or what is the  proper way to get this to work. Multiple websites are going to be using this exchanger server SMTP relay in a secure way. Zero-dimensional ideals and finite-dimensional algebras I have recently begun self-studying Number Theory, and am working on proving:     Show that every integer $n&gt;0$ can be uniquely written as $$n = \sum_{i=0}^mc_i3^i$$ where $c_i \in \{ -1,0,1\}$ and $c_m \neq 0$.   I believe I have already shown the existence portion of this proof correctly, and now I am looking for some hints on the uniqueness part. So far what I have tried is:     Suppose for the sake of contradiction that there is an alternate representation of $n$ besides $n = \sum_{i=0}^mc_i3^i$, say $n = \sum_{i=0}^pb_i3^i$ where we still have $b_p \neq 0$ and $b_i \in \{-1,0,1\}$.    At this point I felt that I first need to establish that $m=p$, and second show $c_i=b_i$ for each $i \in \{1,2,\dots,  m\}$. To show the former, I left off in my proof by contradiction with      Without loss of generality suppose $p&gt;m$. We know there is an integer $q$ such that $m+q=p$. We have two ways of writing $n$, which means $$ \sum_{i=0}^pb_i3^i-\sum_{i=0}^mc_i3^i=0 \\ \implies \left(\sum_{i=0}^mb_i3^i+\sum_{i=m+1}^{m+q}b_i3^i\right)-\sum_{i=0}^mc_i3^i=0 \\ \implies \sum_{i=0}^m(b_i-c_i)3^i+\sum_{i=m+1}^{m+q}b_i3^i=0 \\ \implies \sum_{i=m+1}^{m+q}b_i3^i=\sum_{i=0}^m(c_i-b_i)3^i$$   At this point I am stuck. If anyone has any hints to help me find the contradiction or feel that there is a better way to establish uniqueness, please let me know! Lost my old passport for Schengen visa application Now the cubs are growing, fed on mom's milk, but mom herself hasn't eaten in five months, and is getting thinner by the day.   What does in mean here, and how would the meaning of that piece change if for were used? I do not have enough time to bake my bread.  Can I refrigerate it or let it sit on the counter and bake it later in the day? How to exclude a column from SELECT query? How did Luke know where to land on Dagobah? Can more than one antibody bind the same antigen I'd like to clear something up about antibodies that I'm not sure I've understood in the articles I've read. Looking at concepts such as "affinity maturation", "monovalent antigens" and "polyvalent antigens" it seems as if there are multiple antibodies which can bind to the same antigen. (Although I think I'm right in saying that individual antibodies can only bind to one antigen?) However, when I read sentences like "It has been estimated that humans generate about 10 billion different antibodies, each capable of binding a distinct epitope of an antigen" on the Wikipedia page (for antibody) it does make it sound like it's one antibody per antigen? Is this the case? Or is it one antibody per epitope. Either way, to put this question in a nutshell:  Is it true to say that the same substance could bind with (and, therefore, be recognised by) more than one antibody, or not?  Also, if the answer is no, does that mean that antibodies with different binding affinities are classed as the same antibody or not?