The problem is the following one:  Let $f: A \rightarrow B$ a ring isomorphism. If A is a field, is B a field?  I think It's false, but I can't think of a counterexample, and if it's true I have no idea of how to prove it.
Let $R \simeq S$ be isomorphic commutative rings with unity. Prove the following:  a). If $R$ is an integral domain then $S$ is an integral domain.  For this, I said that If I let $f: R \to S$ be an isomorphism and $(R, + , \circ), (S, \oplus, \ast)$, then let $x, y\in R$,  such that $x \circ y = 0_R$, because $f$ is an isomorphism, $f(x \circ y) = f(x) \ast f(y) = f(0_R)$. If I let $R$ be an integral domain then, as per the definition $R$ is commutative, has unity, and no zero divisors.   So $f(x \circ y) = f(x) \ast f(y) = f(0_R)$ where we have either $f(x \circ y) = f(0_R) \ast f(y) = f(0_R)$ or $f(x \circ y) = f(x) \ast f(0_R) = f(0_R)$.   Since $f$ is an isomomorphism, let $s = f(x), t = f(y)| s,t \in S$. Then $s \ast t = f(0_R)$. By definition, $S$ also has unity so $0_S$ exists. If we map $0_R \to 0_S$. Then either $s = 0_S$ or $t = 0_S$.   So then, since $S$ has unity, is commutative, and has a no zero divisors as we showed above, it is also an Integral domain.  I'm not sure I have this correctly.  b). If $R$ is a field then $S$ is a field. I don't really understand how to do this one. Since $S$ is an integral domain, it has a $0_S$. So I need to show that $S$ distributes $\ast$ over $\oplus$.  Any help with these is appreciated
The scenario:  In a bash script, I have to check if a password given by a user is a valid user password.   I.e suppose I have a user A with password PA.. In the script I asked user A to enter his password, So how to check if the string entered is really his password?... Find three real orthogonal matrices of order $3$ having all integer entries.   I have no idea to solve the problem. I don't know how to start. If $A$ be such matrix then $AA^T=A^TA=I_3$. Please help me. Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$  So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called $a_{n}$ and the denominator be $b_{n}$ Is there a way to use this statement so that I could force the original sequence into the form of $1/\left(1+\frac{1}{n}\right)^n$ How do I change the Google Play country? Can someone tell me how to change the country preference for Google Play? I have searched high and low but in most of the forums people ask you to go to Gmail settings -> edit profile and change the country there. But there is no place to change the country in settings! There is a something called place (which can be marked as current) but that has no impact on Google Play.  I don't want to circumvent country restrictions, I want to permanently change the country. So it's NOT a duplicate of . Query without having to specify table's schema I imported a bunch of tables from SQL Server 2000 to my 2008 database. All the imported tables are prefixed with my username eg: erpadmin.tablename.  In the table properties it lists 'erpadmin' as the db schema. When I write a query I now have to include 'erpadmin.' in front of all the table names which is confusing.  Current result:  select * from erpadmin.tablename   Desired result:  select * from  tablename I want to include some Cyrillic with a TeX Gyre Pagella text and math font document. I'm using TeXLive 2019.  As indicated in , running pdflatex on the following source works just fine:  \documentclass{article}  \usepackage[T1,T2A]{fontenc} \usepackage[russian,main=english]{babel} \usepackage{paratype} \usepackage{tgpagella} \usepackage{substitutefont} \substitutefont{T2A}{\rmdefault}{PTSerif-TLF}  \usepackage{amsmath}  \usepackage[math]{blindtext}  \begin{document}  The topologist is {\Russian Алекс\'{а}ндров}.  \blindmathpaper  \end{document}   However, running xelatex on the following surce produces output from which the indicated Cyrillic just disappears:  \documentclass{article}  \usepackage[T2A,T1]{fontenc} \usepackage[russian,main=english]{babel} \usepackage{amsmath} % load here to avoid \dddot clash \usepackage{fontspec} \usepackage{paratype} \setmainfont[Ligatures=TeX]{TeX Gyre Pagella} \usepackage{substitutefont}% \substitutefont{T2A}{\rmdefault}{PTSerif-TLF}%  \usepackage{unicode-math} \setmathfont{TeX Gyre Pagella Math}  \usepackage[math]{blindtext}  \begin{document}  The topologist is {\Russian Алекс\'{а}ндров}.  \blindmathpaper  \end{document}   Question: How should this second source be fixed so as to allow the Cyrillic to appear, too?  (I did put Map paratype-truetype.map into my local updmap.cfg file and did run updmap-user.) Detecting structured data in Google Webmaster Tools When will my device get the Android 5.1 update (Lollipop)? Android 5.1 (Lollipop) was officially announced on March 9th, 2015.  You can review the high-level changes at the announcement on the official .  Of course, not all devices will get this update right away and others won't get it at all. Each manufacturer and carrier often choose to add their custom modifications which tend to delay the official release.  When will each device get Android 5.1, either officially or unofficially, if ever?  See also: Allow diamond moderators to reverse question migrations? It has happened on occasion, due to the high amount of 10K moderators on SO that questions are migrated to the wrong site, and needs to be reversed. Although currently it is possible to re-open the question on the original site and delete the one on the destination site, it still leave the migrated too banner behind.  It would be useful if diamond moderators could reverse a migration, removing it from the destination site completely and re-activating it on the original site and at the same time removing the banners.  Previously it has been discussed educating the users around when to move questions, however this doesn't stop the problem.  As per Shog9  I have to admit I do like Shog9's suggestion regarding the post being closed as not programming related rather then re-opened, however, reasons for closing it on other sites should also be kept in mind, as this also applies to stuff being sent back forth between the sites. How do you find the Lie algebra of a Lie group (in practice)? Given a Lie group, how are you meant to find its Lie algebra?  The Lie algebra of a Lie group is the set of all the left invariant vector fields, but how would you determine them?  My group is the set of all affine maps $x \rightarrow A.x+v$ from $R^n$ to $R^n$ under function composition. I am attempting to settle a friendly bet.  Would a pendulum swing indefinitely in a hypothetical vacuum (i.e. no air resistance) having a hypothetical frictionless bearing (i.e. no energy lost due to friction) assuming the following   The frictionless vacuum is on Earth (9.8 m/s^2). The pendulum is already in motion and no other external forces other than gravity act on the pendulum. Is there any "superlogarithm" or something to solve $x^x$? Is there any "superlogarithm" or something to solve an equation like this:  $$x^x = 10?$$ In this MWE, I want to obtain bold letter for the letter "H" (compare the two "H") :  \documentclass[10 pt]{beamer} \usetheme{default} \usepackage{lmodern} % load a font with all the characters \usepackage{fontspec} \usepackage{cmbright} \SetSymbolFont{largesymbols}{normal}{OMX}{iwona}{m}{n} \usepackage{amsmath}%loads amsmath that should go before unicode-math \usefonttheme{professionalfonts} \defaultfontfeatures{Mapping=tex-text}  % This seems to be important for mapping glyphs properly  \begin{document} \begin{frame} \[ \mbox{First bold H : }\|x\|_{\mathbf{H}},\; \mbox{Second bold H : }\mathbf{H}  \] \end{frame} \end{document}   but I don't know how doing it. THe "H" produced by this MWE is tiny but no bold effect. If I use      \SetSymbolFont{largesymbols}{normal}{OMX}{iwona}{m}{n}   I obtain bold tiny "H" but the font used still not the same. I want to create a function that performs a function passed by parameter on a set of data. How do you pass a function as a parameter in C? What is "angle of view" in photography? I read on the :     Focal-length determines the angle-of-view seen through a lens for a   given sensor-size. With a full-frame sensor a lens gives the same   angle-of-view as it would on a 35mm film camera. With a smaller   sensor, the angle-of-view becomes smaller. The crop-factor, also   called FLM, is the ratio representing the difference in equivalent   focal-lengths. So a 150mm on a full-frame DSLR such as the Nikon D700   gives the same angle-of-view as a 100mm on a D7000 since its FLM is   1.5X.      Short focal-lengths show a greater angle-of-view compared to longer   ones. [...]   In layman's terms, what is angle-of-view? Is it the same as focal length? If not, how is it different? How is it used? Why do I need to know about it? Comment font sizes are inconsistent on Android An issue similar to that described in  and likely to be related.  The comments for a question may vary in size:         Chrome for Android on Nexus 7 running 4.2.2 Jelly Bean. This also occurs on Firefox for Android: How can I save application settings in a Windows Forms application? How do I play Minecraft with people online, besides using a server? I have two tensors that i must calculate double dot product. matrix A is rank 2  and matrix B is rank 4. I want to multiply them with Matlab and I know in Matlab it becomes: A : B = trace (A*B) but it has one error and it says: Inner matrix dimensions must agree So how can I solve this problem? anybody help me? Thanks Why doesn't an Environment texture appear when the world has a Volume Scatter shader? I'm working on lighting a scene in which I need to have both an Environment texture for the background and Volume Scatter for light cones, but when I check the volume section in World setting and choose the Volume Scatter, the environment becomes black background (but the lighting works as I want).  I'm using Blender 2.72  How can this be fixed?