Let $A$ be a nonzero commutative ring with $1$, and $n$ a positive integer. If $b_1,\ldots,b_n\in A^n$ generate $A^n$ as an $A$-module, is it true that $b_1,\ldots,b_n$ are linearly independent over $A$ (hence a basis)? I suspect this is true (It is true if $n=1$), but I can't seem to prove it.  I guess an equivalent way of asking this question is: if a linear map $A^n\to A^n$ is surjective, is it necessarily injective?  Any help is welcome!
Let $L$ be a finitely generated free module over a commutative ring $A$. Set $n=\operatorname{rank} L$. Let $x_1,\dots,x_m$ be generators of $L$. Then $m \ge n$? If $m = n$, then is $x_1,\dots,x_m$ a basis of $L$?
Children’s books, one in space setting, the other in a jungle Can others in my party go through TSA PreCheck? If I have  (i.e. I have a KTN from TSA and the "pre-check" logo gets printed on my boarding pass) and a travel companion does not, can they go with me through the PreCheck security lines? This is assuming that the travel companion is booked in the same reservation on the same flights and is also a U.S. citizen and not on any no-fly or otherwise increased scrutiny lists.  I'm guessing the answer is no, but I haven't seen anything from TSA specifically addressing this. Do plugin files have to follow a specific convention to be "picked up" by WordPress? How do I put big brackets under different parts of an equation so that I can write, e.g., text specific to these parts? For example: f =  x^3 +  2     |___|  |__|        |     |    text 1  text 2 How to change certain pages into landscape/portrait mode How can I change some pages into document into landscape mode, while leaving the others in portrait mode (or vice versa)? What's interesting in latus rectum? I'm a maths teacher in Italian secondary school and I've been spending some time trying to construct "meaningful" problems about conic sections. I particularly like problems which focus on practical applications of the curves (in engineering, physics, finance, etc.), on their geometric properties, or both.  I know there's this really easy formula to find the latus rectum of a conic from its key parameters. It's quite nice and straightforward to derive it, but...  ... What's its use? Why should anyone be interested in knowing the latus rectum of a curve? Is there any physical phenomenon, any architectural trick, or anything else where its position and/or length plays a key role?   I know just the relation between angular momentum, standard gravitational parameter and latus rectum in an astronomical orbit ($l=\frac{h^2}{GM}$, where $l$ is half the latus rectum, and $h$ is the ratio between the angular momentum associated to the orbiting body and its mass). It's cool, but a bit tricky, and probably too far-fetched for secondary school students... Is there anything simpler? Or just anything else, at any difficulty level?  Many thanks! Solving an equation over the reals: $ x^3 + 1 = 2\sqrt[3]{{2x - 1}}$ Why does node not evaluate Math.tan(Math.PI/2) to Infinity but Chrome V8 does? How I can create a graph of acceleration and speed - tikz? How can I create a graph of acceleration and speed like the one in the picture. Knowing that they are each other's consequence?    \documentclass{article} \usepackage{tikz}  \begin{document}  \begin{tikzpicture}               \end{tikzpicture} \end{document} Prove that if prime $p$ divide $a_{2k}-2$, then $p$ divide also $a_{2k+1}-1$. In my  I was printing a double using cout that got rounded when I wasn't expecting it.  How can I make cout print a double using full precision? Center of a group ring? Let $R$ be a ring with unity and let $G$ be a group. Then what is the center of the group ring $R(G)$? I feel that the center $Z(R(G))$ is $R(Z(G))$. I assumed if $x=\sum_{i=1}^{n} r_i g_i$, then it should commute with every element of $R(G)$. In particular it should commute with $g=1 \cdot g \in R(G)$ Thus we get $$\big(\sum_{i=1}^{n} r_i g_i\big)g=g\big(\sum_{i=1}^{n} r_i g_i\big)$$ Thus I get $$g_i \cdot g=g_i \cdot g \text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ } \forall g \in G$$ Hence $x \in Z(R(G)) $. Is this correct? If I play a spell with {X} in the mana cost, and I have a card out that reduces the casting cost by 2 generic mana, what happens to the X? Does it start at -2?  For example: I have  in play. If I play  for just 1 blue mana, do we draw 2 cards? Proof of uniform limit of Continuous Functions The uniform limit of Continuous Functions is continuous.   Proof Let $\varepsilon &gt; 0$. There exists $N$ in natural numbers such that $n&gt;N$ implies  $|f_n(x)-f(x)| &lt; \frac{\varepsilon}{3}$ for all $x$ in $S$. In particular $|f_{N+1}(x)-f(x)| &lt; \frac{\varepsilon}{3}$ for all $x$ in $S$. Since $f_{N+1}$ is continuous at $x_0$ there is a $\delta&gt;0$ such that, $x$ in $S$ and $|x-x_0| &lt; \delta$ imply $|f_{N+1}(x)-f_{N+1}(x_0)|&lt; \frac{\varepsilon}{3}$ Now we conclude $x$ in $S$ and $|x-x_0| &lt; \delta$ imply $|f(x)-f(x_0)|&lt; 3 \cdot \frac{\varepsilon}{3} = \varepsilon$.  Can someone explain what is happening here? I don't follow it all. Huge paged pool probably caused by fltMgr.sys , what to do? Find the radius of the inscribed circle of a right triangle. Find the radius of the inscribed circle of a right triangle. The triangle's height is $\sqrt{6} + \sqrt{2}$ while the bisector of the right angle is 4.  This seemed like a generic similar triangles problem but it isn't that simple when I tried to solve it. Any help?  EDIT: Here is a picture of the problem When to use "I" and "me" in an answer Strange folder in my home folder after a failed run of BleachBit Is there a specific term for the association of certain verb(s) with certain nouns (as direct objects)? Or vice versa. Or perhaps a name for the pair itself?  For example:   hold an opinion make a complaint offer an apology   Also, where might I find a list of such pairs?    EDIT: I chose my title wording randomly. I have no idea whether 'cognate verb' is the correct term for these things. However, I've just found out that 'cognate objects' are a defined class in linguistics, so perhaps my choice of words was inappropriate. I'm going to leave the title as it is, anyway. Usage and origin of "sister" in expressions like "sister company, sister ship, sister site" etc