R is a quasi-semi-local ring, I is ideal of R. Prove Jac(R/I)=(JacR + I)/I R is a quasi-semi-local ring (R has finitely many maximal ideals), I is ideal of R. Prove Jac(R/I)=(JacR + I)/I, in which Jac is Jacobson radical.   It's a problem in my teacher's book. I think we should divide maximal ideals of R into $M_1,...,M_k$ and $M_{k+1},...,M_n$, the first set consists of maximal ideals containing I. But I don't know what to do next
Image of Jacobson Radical is the Jacobson Radical I'm stuck on the following problem:      Let $R$ be a commutative ring with $1$, and suppose $R$ is semilocal with maximal ideals $\mathfrak m_1, ... , \mathfrak m_s$.  Let $\mathfrak a$ be an ideal of $R$. The problem is to show that under the usual surjection $\pi:R \rightarrow R/\mathfrak a$, the Jacobson radical of $R$ is sent to the Jacobson radical of $R/\mathfrak a$.   My attempt: suppose that $\mathfrak m_1, ... , \mathfrak m_t$ contain $\mathfrak a$, but $\mathfrak m_{t+1}, ... , \mathfrak m_s$ do not.  Then $Jac(R/\mathfrak a) = (\mathfrak m_1/ \mathfrak a) \cap \cdots \cap (\mathfrak m_t/ \mathfrak a)$.  Obviously $\pi Jac(R) \subseteq Jac(R/ \mathfrak a)$, and the converse inclusion is equivalent to the following claim: if $x \in \mathfrak m_1 \cap \cdots \cap \mathfrak m_t$, then there exists a $y \in \mathfrak m_1 \cap \cdots \cap \mathfrak m_s$ such that $x - y \in \mathfrak a$.    This should be some application of the Chinese remainder theorem, but I'm not seeing it.  All I have to go on is that there exist $m_i \in \mathfrak m_i$ ($t+1 \leq i \leq s$) with $m_i \not\in I$.  I also haven't found out a way to use the fact that $\mathfrak m_1 ,... , \mathfrak m_s$ are all the maximal ideals of $R$.  I was also thinking about looking at some explicit maps.  If we let $\Gamma$ be the composition $$\frac{(R/\mathfrak a)}{Jac(R/ \mathfrak a)} \xrightarrow{\cong} \frac{(R/\mathfrak a)}{(\mathfrak m_1/\mathfrak a)} \oplus \cdots \oplus \frac{(R/\mathfrak a)}{(\mathfrak m_t/\mathfrak a)} \xrightarrow{\cong} R/\mathfrak m_1 \oplus \cdots \oplus R/\mathfrak m_t$$ then we have a commutative diagram $$\begin{matrix} R/Jac(R) &amp;  \rightarrow &amp; \frac{(R/\mathfrak a)}{Jac(R/\mathfrak a)} \\ \downarrow &amp; &amp; \downarrow \Gamma \\ R/\mathfrak m_1 \oplus \cdots R/\mathfrak m_s &amp; \rightarrow &amp; R/\mathfrak m_1 \oplus \cdots \oplus R/\mathfrak m_t \end{matrix}$$ where the vertical arrows are isomorphisms, and the horizontal arrows are surjections.
I'm trying to determine the best way to truncate or drop extra decimal places in SQL without rounding.  For example:  declare @value decimal(18,2)  set @value = 123.456   This will automatically round @value to be 123.46, which is good in most cases.  However, for this project, I don't need that.  Is there a simple way to truncate the decimals I don't need?  I know I can use the left() function and convert back to a decimal.  Are there any other ways? Connecting Asus USB-N13 Wireless Adapter I'm trying to connect my desktop (running 64-bit Ubuntu 12.04) to an Airport Extreme Base Station by using an ASUS USB-N13 wireless adapter.  I plugged it in, turned on the desktop, and the adapter was immediately recognized by the system. I found my home network listed in the available networks, typed my password (with the correct encryption) and hit connect. After that, the wireless icon just flashes (propagates? emits?) indefinitely, and every couple of minutes the "Wireless Authentication Required" box pops back up.   I don't know what to do. I've seen some solutions talking about an "rt2xxx" or "rt3xxx" driver or chipset. I don't know exactly what it is, but when I typed lsmod | grep rt I got:  parport_pc rtl8192cu rtl8192c_common rtlwifi mac80211 cfg80211 parport   I'm pretty much lost. Please AskUbuntu, what do? Selecting attributes or rows of polygons that intersect two attributes of overlaying polygon using ArcGIS Pro When & How often do 4e Monster powers recharge? How often should I roll power recharge rolls and at which point in the monster's turn do I re-roll them?  I'm trying to find the official definition of the recharge mechanic in the online compendium and I'm not seeing it there. Enable Wireless Monitor mode for traffic sniffing I need to sniff all the wifi traffic that I can capture using wireshark in Ubuntu. My wireless interface is named as eth1 in the interfaces list. When I select that interface in Wireshark and start new sniff, I got following error;    As  thread says, I tried to installed the mentioned driver but I got an error in installation saying that that driver is not supported by my device. After that I tried the second answer in the same thread and run following command to enable monitor mode in my wireless card.  echo 1 &gt; /proc/brcm_monitor0   Then I saw a new Ethernet interface (not a wireless interface ) called prism0 in wireshark interface list. I couldn't start a sniff using that interface using monitor mode because in that interface settings, monitor mode check box has been disabled.  I have installed a broadcom wireless driver in Additional Drivers as follows:    Is there an error listing my Wifi interface as eth1 rather than wlan0 in the interfaces list? How can I start my wireless interface in monitor mode for sniffing?   Thank you. I found this heart while exploring the crimson biome - what is it? Multiple solutions for a particle in an infinite potential well I am starting to learn about quantum mechanics and was wondering about the solutions to a question about a particle in an infinite potential well of length $L$.  The solutions are: $$\psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{\pi n x}{L} \right)$$  And I was wondering what is the meaning of $n$. Isn't there supposed to be only one wave function? In that case, $\Psi = \sum_{n=1} ^\infty a_n \psi_n$, but what are the $a_n$? Or maybe it is probabilistic, and we don't know which $\psi_n$ the particle is currently in? Proof that a sequence of continuous functions $(f_n)$ cannot converge pointwise to $1_\mathbb{Q}$ on $[0,1]$ Central Moments of Symmetric Distributions I am trying to show that the central moment of a symmetric distribution:     $${\bf f}_x{\bf (a+x)} = {\bf f}_x{\bf(a-x)}$$ is zero for odd numbers. So for instance the third central moment $${\bf E[(X-u)^3] = 0}.$$ I started by trying to show that $${\bf E[(X-u)^3] = E[X^3] -3uE[X^2] + 3u^2E[X] - u^3}.$$ I am not sure where to go from here, any suggestions? Is there a better way to go about proving this? iPhone call log / history Last activity name still missing from migrated question According to this post  this should be fixed. But it happened again.  See Image below (link to )  There was also another migrated question () in the list where the name was missing, but it was edited on meta. I would like to update VirtualBox on my 32 bit Ubuntu 15.04 machine from 4.3.26 to 4.3.28, using a Debian package I manually downloaded from . But simply double-clicking the package opens up the Software Centre which gives an error that installing it would break an existing package (the exact message was:   Breaks existing package 'virtualbox' conflict: virtualbox-4.3 ()   and it does not appear to have an option to ignore this and proceed with the installation anyway. Is there a command-line work-around this problem, perhaps? How to strike through obliquely, e.g. to indicate cancellation? How can I strike through an expression (e.g. "2x") obliquely, e.g. from bottom left to top right, so as to indicate cancellation? With the \sout command from the ulem package, I can only strike horizontally. Removing unnecessary vertices? I have over sampled polygons that I need to simplify, without changing the shape. basically, I want to reduce the number of vertices, by eliminating the ones that are useless and have the polygons represented with the significant vertices only. I used to do that with ET GeoWizards Plug-in, but I don't have a license anymore. My table has 500.000 features, water polygons.   Any ideas? 'No Such File or Directory" Error When Working with Shared File I'm attempting to open a file on a Linux computer through file sharing. The file is stored on a Macbook Pro and both computers are running Blender 2.77. However, when I try to open the file on the Linux computer, I get an error of, "No Such File or Directory." I don't think this is a file permissions error as my permissions settings are set so that "Everyone" can read and write my file. What can I do to solve this? How can I display "recent folders" in the sidebar of "Save as" dialogue? I was using 14.04 and upgraded to 16.04 yesterday.   In 14.04 I had recent folders on top of the sidebar of the Save as dialogue when I saved something from the net (Firefox if that helps). But that feature disappeared in 16.04 and I really want to have it back.   Here is a picture of what I'm trying to explain:  (the save as dialogue in 16.04 without the recent folders on top) I've been trying to develop an intuition based understanding of Bayes' theorem in terms of the prior, posterior, likelihood and marginal probability. For that I use the following equation: $$P(B|A) = \frac{P(A|B)P(B)}{P(A)}$$ where $A$ represents a hypothesis or belief and $B$ represents data or evidence. I've understood the concept of the posterior - it's a unifying entity that combines the prior belief and the likelihood of an event. What I don't understand is what does the likelihood signify? And why is the marginal probability in the denominator? After reviewing a couple of resources I came across this quote:     The likelihood is the weight of event $B$ given by the occurrence of $A$ ... $P(B|A)$ is the posterior probability of event $B$ , given that event $A$ has occurred.   The above 2 statements seem identical to me, just written in different ways. Can anyone please explain the difference between the two? Windows 10 non-persistent installation (live ISO) We've got a Windows 10 installation running in an environment where it can occasionally lose power and restart. However, this sometimes leads to corruption of files.  I've had a look at Windows "To-Go" but this seems to be persistent. And I've also checked out virtualisation, but that would require us to run a non-persistent Windows 10, in another Windows 10 environment.  Is it possible to install Windows 10 as a "non-persistent live CD"? So that a user will have a seemingly fresh Windows 10, but doesn't require to re-install it at each startup? Adding objects algorithm in Objective-C I can read the MySQL documentation and it's pretty clear. But, how does one decide which character set to use? On what data does collation have an effect?  I'm asking for an explanation of the two and how to choose them.