Let $A$ a finite ring. Prove that $A$ is a field, or $A$ has zero divisors.    I begin to assume that $A$ has no zero divisors but I don't know continue... \  How would be this proof? thanks! :)
Let $R$ be a finite ring with unity. Prove that every nonzero element of $R$ is either a unit or a zero-divisor.
How long does the ubuntu installation take and what to do after it? Crash course on algebraic geometry with view to applications in physics Could you please recommend any good texts on algebraic geometry (just over the complex numbers rather than arbitrary fields) and on complex geometry including Kahler manifolds that could serve as an informal introduction to the subject for a theoretical physicist (having in mind the applications in physics, e.g. in the string theory)?   What I want for a moment is to get some informal picture of the subject rather than being dug up into the gory details of the proofs and lost in higher and higher layers of abstraction of commutative algebra and category theory. The texts I have found so far are all rather dry and almost completely lack this informal streak, and all of them are geared towards pure mathematicians, so if there exists something like "Algebraic geometry for physicists" and "Kahler manifolds for physicists" (of course, they would probably have different titles :)), I would greatly appreciate the relevant references. how much money it takes for a day without including accommodation In Uk Spacetime Torsion, the Spin tensor, and intrinsic spin in Einstein-Cartan theory In  gravity, the action is the usual  but now the Torsion tensor is allowed to vary as well (in usual GR, it is just set to zero).  Variation with respect to the metric gives:  $$R_{ab}-\frac{1}{2}R g_{ab}=\kappa P_{ab} \quad (1)$$  where $P_{ab}$ is the canonical stress energy tensor.  Variation with respect to the torsion tensor ${T^{ab}}_c$ gives:  $${T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d = \kappa {\sigma_{ab}}^c \quad (2)$$  where ${\sigma_{ab}}^c$ is the .  By contracting that equation, I can see that if the spin tensor is zero, the Torsion tensor is identically zero:  $${T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d = \kappa {\sigma_{ab}}^c = 0$$ $$ {g^b}_c({T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d) = 0$$ $$ {T_{ab}}^b + {g_a}^b{T_{bd}}^d - {g_b}^b {T_{ad}}^d = 0$$ $$ {T_{ab}}^b + {T_{ad}}^d - 4 {T_{ad}}^d = 0$$ $$ {T_{ad}}^d = 0$$ $${T_{ab}}^c + {g_a}^c{T_{bd}}^d - {g_b}^c {T_{ad}}^d = 0 \quad \Rightarrow \quad {T_{ab}}^c =0$$  My understanding is that:  The spin tensor and the stress energy tensor, are defined entirely in terms of whatever matter Lagrangian we add to the theory.  Therefore, from above, the equations in vacuum are exactly the same as normal GR (so solving for outside matter, only the boundary conditions with the matter could be different).  Assuming my understanding up to here is correct, my line of questioning is:   How is the Spin tensor (and hence Torsion) related to the concept of material with intrinsic spin? Hopefully answered by 1, but if the matter has zero intrinsic spin, yet we have an extended "spinning" body, is the spin tensor still zero (as I'd consider that orbital angular momentum then)? Does this mean Einstein-Cartan predictions are identical to normal Einstein GR if (and only if) the instrinsic spin of any particles and fields in the theory are zero? align xticks in PGFPlots Plotting in the Complex Plane How do I decompile a .NET EXE into readable C# source code? I wrote a C# application for a client a couple of years ago, but I no longer have the source code.  All I have is the EXE that I deployed on the client's PC.  Is there a way I can generate C# source code from the EXE? Novel about a dragon lord who has to wear gloves at all times Output the common lines (similarities) of two text files (the opposite of diff)? When starring a message from the star board, it's not reflected in the main chat window How to compare the strings using < (Greater than symbol) Below is my input file:  PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:16:46.523.177, 2 PS Sensor Value = -5.501000 , Min = -5.583000 , Max = -5.319000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:16:46.523.210, 3 PS Sensor Value = 15.996000 , Min = 15.814000 , Max = 16.078000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:16:46.523.231, 4 PS Sensor Value = -16.505000 , Min = -16.587000 , Max = -16.323000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:16:46.523.263, 5 PS Sensor Value = 6.509000 , Min = 6.327000 , Max = 6.591000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:16:46.523.302, 6 PS Sensor Value = 4.002000 , Min = 3.820000 , Max = 4.084000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:17:46.481.557, 1 PS Sensor Value = 6.199000 , Min = 6.017000 , Max = 6.281000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:17:46.518.691, 2 PS Sensor Value = -5.503000 , Min = -5.585000 , Max = -5.321000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:17:46.523.156, 3 PS Sensor Value = 15.996000 , Min = 15.814000 , Max = 16.078000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:17:46.523.195, 4 PS Sensor Value = -16.505000 , Min = -16.587000 , Max = -16.323000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:17:46.523.221, 5 PS Sensor Value = 6.509000 , Min = 6.327000 , Max = 6.591000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:17:46.523.240, 6 PS Sensor Value = 4.002000 , Min = 3.820000 , Max = 4.084000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:18:46.480.644, 1 PS Sensor Value = 6.199000 , Min = 6.017000 , Max = 6.281000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:18:46.522.615, 2 PS Sensor Value = -5.501000 , Min = -5.583000 , Max = -5.319000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:18:46.522.729, 3 PS Sensor Value = 15.996000 , Min = 15.814000 , Max = 16.078000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:18:46.522.765, 4 PS Sensor Value = -16.505000 , Min = -16.587000 , Max = -16.323000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:18:46.522.788, 5 PS Sensor Value = 6.509000 , Min = 6.327000 , Max = 6.591000 PA43410-2,1,3,/vobs/atlas-idc/src/utils/logger/IDCLogger/IDC.cpp,48,19:18:46.522.810, 6 PS Sensor Value = 4.002000 , Min = 3.820000 , Max = 4.084000   I need to compare the PS Sensor Value, Min and Max value. (Min greater than PS Sensor greater than Max)   I'm expecting the output like below and here I should have only the normal lines.  What is normal lines? :  If Min value greater than PS Sensor value &amp;&amp; PS Sensor value  is greater than Max value then the line is normal lines and that should removed.        Expected Output:                                                                                                                                    PA43410-2  PS Sensor Value = 6.509000 , Min = 6.327000 , Max = 6.591000 PA43410-2 PS Sensor Value = 6.199000 , Min = 6.017000 , Max = 6.281000  Eg:     If   Min &lt; PS sensor value &lt; Max,    // Don’t care the normal  lines.          Throw this line away.    Else           Pull this line to consolidated new file. //Only focus on abnormal lines. What's the difference between absolute path &amp; relative path when using any web server or Tomcat? Why are there $736$ matrices $M\in \mathcal M_2(\mathbb{Z}_{26})$ for which it holds that $M=M^{-1}$? I'm currently trying to introduce myself to cryptography.  I'm reading about the Hill Cipher currently in the book Applied Abstract Algebra. The Hill Cipher uses an invertible matrix $M$ for encryption and the inverse matrix $M^{-1}$ for decryption. The book recommends to use a matrix $M$ for which it holds that $M=M^{-1}$, since we then have the same key for encryption and decryption.  In the book, they use a $2 \times 2$ matrix $M$ over $\mathbb{Z}_{26}$ as example, and state that for there are 736 $2 \times 2$ matrices for which it hold that $M=M^{-1}$.  I'm trying to pick up on as much as possible when reading things, since I find it counter-productive for learning to skip something, when you don't get the theory behind it. Can someone enlighten to me, as to why it is that there are 736 possible $2 \times 2$ $M=M^{-1}$ matrices and how to find them? Profile Gravatar get change Do not know whether someone else observe it or not but today I found my profile Gravatar changed. It is now different than earlier I have. Is it because today I have updated my profile information?        My previous gravatar looks like: Should I remove Adult Dependent from my 2019 tax return so they will receive a COVID Stimulus Check? On my 2019 Tax Return I claimed my 28 year old girlfriend as a dependent. This makes her ineligible to receive a COVID Stimulus Check. My return was already accepted and I have already received my refund. Should I amend my return, remove her as a dependent? Would she then be eligible to receive the stimulus check? The basic concept of Quotient Group is often a confusing thing for me,I mean can any one tell the intuitive concept and the necessity of the Quotient group, I thought that it would be nice to ask as any basic undergraduate can learn the intuition seeing the question. My Question is :   Why is the name Quotient Group kept,normally in the case of division, let us take the example of $\large \frac{16}{4}$ the Quotient of the Division is '$4$' which means that there are four '$4$'s in $16$, I mean we can find only $4$ elements with value $4$     So how can we apply the same logic in the case of Quotient Groups,like consider the Group $A$ and normal subgroup $B$ of $A$, So if $A/B$ refers to "Quotient group", then does it mean:           Are we finding how many copies of $B$ are present in $A$??, like in the case of normal division, or is it something different ??       I understood the Notion of Cosets and Quotient Groups,b ut I want a different Perspective to add Color to the concept. Can anyone tell me the necessity and background for the invention of Quotient Groups?  Note: I tried my level best in formatting and typing with proper protocol,if in case, any errors still persist, I beg everyone to explain the reason of their downvote (if any), so that I can rectify myself, Thank you. How do you prove that a harmonic planar mapping $f(x,y) = u(x,y) + i v(x,y)$ for real $u,v$ can be written as $f(x,y) = \phi(x,y) + \overline{\psi}(x,y)$ where $\phi$ is a holomorphic function, and $\overline{\psi}$ is an anti-holomorphic function (conjugate of a holomorphic function)? New install - no network devices available - Realtek Device b822 I just purchased a Lenovo Yoga 730 15" laptop, and installed Ubuntu 16.04 in place of Windows 10. Install seemingly worked fine, but now I have no wifi device available.  Tried a reinstall just to be sure, and the same issue is happening. I don't believe it to be a hardware issue, as wifi was working without issue on Windows before I wiped it out.  When I run rfkill list I see:  0: ideapad_wlan: Wireless LAN Soft blocked: no Hard blocked: no   But when I run lspci I only see one item in the list claiming to be a network device:  3a:00.0 Network controller: Realtek Semiconductor Co., Ltd. Device b822 (rev ff)   I have a USB wifi adapter and that gets me connected without issue, but obviously that is not the desired solution.  Does anyone have any insight on what might be causing the issue?  Output from modinfo -p rtl8822be:  fix_rate: (int) debug_level: (int) swenc:Set to 1 for software crypto (default 0) (bool) ips:Set to 0 to not use link power save (default 1) (bool) swlps:Set to 1 to use SW control power save (default 0) (bool) fwlps:Set to 1 to use FW control power save (default 1) (bool) msi:Set to 1 to use MSI interrupts mode (default 1) (bool) dma64:Set to 1 to use DMA 64 (default 0) (bool) aspm:Set to 1 to enable ASPM (default 1) (int) debug:Set debug level (0-5) (default 0) debug_mask:Set debug mask (default 0) (ullong) disable_watchdog:Set to 1 to disable the watchdog (default 0) (bool) Can I take a sex-toy on plane through UAE? I will be traveling from Australia to Austria, transferring in Abu Dhabi. Is it alright for me to bring a vibrator in the check-in luggage, or are there laws against that?  It is approximately 17cm long. Are there any precautions to take? This was posted on facebook and people are saying it is incorrect, it should be: "...as you and I"   Which is correct?