Let $A \neq0$ be a ring. Then the following are equivalent  i) A is a field  ii) the only ideals in A are $0$ and $(1)$  iii) every homomorphism of A into a non-zero ring B is injective.   I have shown $\ \ \ i)\rightarrow ii) \ \  $  and $\ \ \ iii)\rightarrow i) \ \  $  But couldnt do $\ \ \ ii)\rightarrow iii) \ \  $  My thoughts: I think it is enough to show that for a homomorphism $\phi: A\rightarrow B,\ \ \ $ $Ker \ \phi=(0)$ but I dont know how can I show  $Ker \ \phi\neq(1)$  which  automatically gives  us $Ker \ \phi=(0)\ $ as    $\ Ker \ \phi$ can only be $(1)$ or $(0)$
Let $R$ be a simple ring and $T$ be a non-zero ring. Let $f\colon R \rightarrow T$ be a ring homomorphism. Show $f$ is injective.  Proof: $\ker f \lhd R$, so $\ker f=R$ or $\ker f=\{0\}$. If $\ker f=R$ then $R/R \cong \{0\}$, but $T \neq \{0\}$, which is a contradiction. I can't see why does the last part means contradiction.   Thanks for any assistance!
How do I remove Windows but keep Ubuntu? Deleting files older than 30 days based on filename as date If I have a folder of files with their filenames as the date they were created:  2019_04_30.txt 2019_04_15.txt 2019_04_10.txt 2019_02_20.txt 2019_01_05.txt   How would I compare the files names against todays current date  $ date "+%Y_%m_%d" &gt;&gt;&gt; 2019_04_30   If the file names date is greater than 30 days then delete it. I would expect to end up with  2019_04_30.txt 2019_04_15.txt 2019_04_10.txt   I don't have to follow this naming convention, I could use a more suitable date format. I am trying to import an FBX file from Unity into Blender. Does anybody know how to do that? I'm really new to Mac OSX and UNIX based systems. I wanted to run a few Windows games on my Mac Mini so I started installing Wine and MacPorts. I think the installs are correct, but when I type sudo port install wine I see a not found error in my terminal. 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Skype, network, languages, sound (also keyboard sound up/down stops working) and date are gone. Restarting unity-panel-service does not help.  Any thoughts on how to resolve these? Drawing a large binary matrix as colored grid in TikZ Why can't the Klein-Gordon equation with a Couloumb potential describe the hydrogen atom?  Why can the first order Dirac equation explain it? What are the failures? Prove : $p$(n│even number of ODD parts)=$p$(n│distinct parts ,number of ODD parts is even ) I'm trying to prove the following Integer Partition claim :   $p$(n│even number of ODD parts) = $p$(n│distinct parts ,number of ODD parts is even) .  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While certain parts seem clear and logical, I can't say I get the whole picture.  I'll try to explain the counter-intuition I get with the classical example of : the probability space $\Omega = \{ HH, HT, TH, TT \}$ and a r.v. $X(\omega)$ equal to the number of heads.  At times $0$, $1$ and $2$ the available information is represented using $\sigma$-algebras $\mathcal{F}_0=\{\emptyset,\Omega\}$, $\mathcal{F}_1=\{\emptyset, \Omega, \{HH,HT\},\{TH,TT\}\}$ and $\mathcal{F}_2=\{\emptyset, \Omega,\{HH,HT\},\{TH,TT\},\{HH\},\{HT\},\{TH\},\{TT\}\}$.  One can notice that $X(\omega)$ is not measurable with respect to $\mathcal{F}_0$ and $\mathcal{F}_1$, because $X^{-1}((\frac{3}{2}; +\infty))=\{HH\}$. To me it is quite surprising: intuitively $X$ makes perfect sense at all times. In particular it has an expected value at time $0$, which I interpret as that the probability and value of all outcomes $\{\omega\}$ can be computed. How do you think of a non-measurable function?  Here's another way of expressing the same confusion. The most natural choice of $\sigma$-algebra in a finite discrete case is $\mathcal{F}=2^\Omega$, and it is implicitly used in all elementary probability problems. However, this choice of $\mathcal{F}$ does not reflect the fact that some information is known or unknown, conditional probability does. Does it mean that the statement "$\sigma$-algebra is known information" make sense only in conditioning? Why is it convenient then? Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$. Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.  So $(a^2+b^2)(c^2+d^2) = a^2c^2+a^2d^2+b^2c^2+b^2d^2$  and $(ac+bd)^2 = a^2c^2+2acbd+b^2d^2$  So the problem is reduced to proving that $a^2d^2+b^2c^2\ge2acbd$ but I am not sure how to show that I've noticed this a few times on the Feed activity on the Android app, but never thought anything of it.  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Remove space before punctuation I need to remove locally spaces arround (especially before) punctuation signs. This spaces are added by french polyglossia.  \documentclass{article} \usepackage{fontspec} \usepackage{polyglossia} \setmainlanguage{french}  \newcommand{\nospace}[1]{\texttt{#1}}  \begin{document} \nospace{a:b} \end{document}     Thanks for help.