Radical of a ring Let $A$ be a commutative ring with unity. Let the radical $\operatorname{Rad}(A)$ of $A$ be the ideal consisting of all nilpotent elements of $A$.      Is $\operatorname{Rad}(A)$ of $A$ the same as the intersection of all maximal ideals of $A$?    I think this should be a standard result so I'd grateful if someone could point me to a reference or a hint on how to prove it will be even better!  Thank you!
Standard example where Jacobson radical not equal nilradical Is there a standard example in commutative algebra of a ring where the jacobson radical does not equal the nilradical?
If you, for example, have to add a person to an application whilst that person is on the phone, how do you politely ask for that person's gender if the voice and/or name has not proven decisive?  To clarify why I posted this question on this site: I was looking for an answer within the context of the English language and usage. The given answers have proven that this question really isn't off-topic, other languages know different distinctions that are relevant to addressing this issue. I have purchased a rather large bag of long-grain white rice (25 lbs.) and need to know the best way for storing it long-term. This rice will be used as both a food-storage food source and as my everyday source of rice for cooking, so the storage solution should (hopefully) be resealable, rather than once-it's-open-it's-open. I eat rice maybe three times a month, so it isn't really "everyday" but it will be my go-to when I have it. What are the badges I can earn on each site, and what are the exact criteria for earning each badge? Using $\text{n}^{\text{th}}$ root of unity  $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$  Prove that   $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$ Top posts taken from hidden communities From Edit Profile &amp; Settings, we have below information  Hide Communities  Choose which communities will appear in the Communities and Top Network Posts sections of your profile Hiding a community means other users will not see it in your Communities list or Top Network Posts. It does not, however, mean your activity is private.   So it is clearly mentioned  Hiding a community means other users will not see it in your Communities list or Top Network Posts.  But, I could see the top network posts are taken from hidden communities for ex: I have marked 'Christianity' as a hidden community in . But, the top posts has a question taken from Christianity community &quot;&quot; Encrypting as3 flash .swf How do I pre-install Ubuntu for someone (OEM install)? I think my MacBook Pro battery was replaced by Apple's service team. How do I check that? Undefined reference to static class member Can anyone explain why following code won't compile? At least on g++ 4.2.4.  And more interesting, why it will compile when I cast MEMBER to int?  #include &lt;vector&gt;  class Foo {   public:       static const int MEMBER = 1;   };  int main(){       vector&lt;int&gt; v;       v.push_back( Foo::MEMBER );       // undefined reference to `Foo::MEMBER'     v.push_back( (int) Foo::MEMBER ); // OK       return 0; } How to make crs transformation of a QgsVectorLayer? I need to transform QgsVectorLayers in python from one crs into another crs. By googling I found the following solution (), which works for points only:  crsSrc = QgsCoordinateReferenceSystem(4326)    # WGS 84 crsDest = QgsCoordinateReferenceSystem(32633)  # WGS 84 / UTM zone 33N xform = QgsCoordinateTransform(crsSrc, crsDest) pt1 = xform.transform(QgsPoint(18,5))   Is there a way to make a crs transformation of an entire layer? Should flawed questions be closed immediately? I can't delete more than 5 of my own posts in a single day. Is this a bug, or intended behavior? I'm trying to clean up old, crappy posts of mine.    Similar-looking problem, but it claims to be : How do "Dummy Variables" work? I do not understand how dummy variables work in math.   Suppose we have:  $$I_1 = \int_{0}^{\infty} e^{-x^2} dx$$  How is this equivalent to:  $$I_2 = \int_{0}^{\infty} e^{-y^2} dy$$  How does this dummy variable system work?   Since $y$ is the dependent variable for $I_1$ How can $y$ itself be and independent variable for $I_2$  ??  Thanks! Finding $\int^{\infty}_{0}\frac{\ln^2(x)}{(1-x^2)^2} dx$ If a function has a finite limit at infinity, does that imply its derivative goes to zero? I've been thinking about this problem: Let $f: (a, +\infty) \to \mathbb{R}$ be a differentiable function such that $\lim\limits_{x \to +\infty} f(x) = L &lt; \infty$. Then must it be the case that $\lim\limits_{x\to +\infty}f'(x) = 0$?  It looks like it's true, but I haven't managed to work out a proof. I came up with this, but it's pretty sketchy:  $$ \begin{align} \lim_{x \to +\infty} f'(x) &amp;= \lim_{x \to +\infty} \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} \\ &amp;= \lim_{h \to 0} \lim_{x \to +\infty} \frac{f(x+h)-f(x)}{h} \\ &amp;= \lim_{h \to 0} \frac1{h} \lim_{x \to +\infty}[f(x+h)-f(x)] \\ &amp;= \lim_{h \to 0} \frac1{h}(L-L) \\ &amp;= \lim_{h \to 0} \frac{0}{h} \\ &amp;= 0 \end{align} $$  In particular, I don't think I can swap the order of the limits just like that. Is this correct, and if it isn't, how can we prove the statement? I know there is  already, but I think this is different in two aspects. First, that question assumes that $\lim\limits_{x \to +\infty}f'(x)$ exists, which I don't. Second, I also wanted to know if interchanging limits is a valid operation in this case. How to export / dump a MySql table into a text file including the field names (aka headers or column names) How to delete files listed in a text file Last digits, numbers Can anyone please help me?  1) Find the last digit of $7^{12345}$   2) Find the last 2 digits of $3^{3^{2014}}$.  Attempt: 1)  By just setting the powers of $7$ we have $7^1 = 7$, $7^2=49$, $7^3=343$, $7^4 = 2401$, $7^5 = 16807$, $7^6 = 117649$, $\dots$  After the power of $4$, the last digits will repeat. Then by noticing the pattern the digits will end in $7,9,3$ and $1$.  Then we can divide the exponent $(12345)$ by $4$ since this is the cycle that makes it repeat. Then $12345 : 4$ has remainder $1$. So $7^1 = 7$ is the unit digit to $7^{12345}$.   So the last digit is $7$.   I know how to do it like this, the problem does not state how to find the last digit, but I know it has something do do with Euler's theorem.  for part $2$) I don't know how to start. Can anyone please help me?   Thank you for the help. YA dystopia novel of battle school with unusual military control scheme Some sort of battle school (no, not Ender's Game), where teens are being trained as military commanders.   The control system for their army hinges around some sort of system that requires extremely fine motor control -- being able to "push" around digital representations of military units using a special interface that interprets minute finger movements.  The boy protagonist eventually meets a girl in the battle school, and is astonished to learn that most people do not live as well as he does -- for instance, he gets a fresh jumpsuit outfit every day and disposes of the old one; this is a luxury.  One scene I recall is them kissing for a moment in some sort of underground passageway; the boy protagonist notes the exact length of the kiss, as he has been trained to have heightened time perception.  I don't recall much about other commander characters; I think they have different special abilities, and one other major character might have physical deformities, possibly because of postwar-radiation-related birth defects.  They have been training for an upcoming, prearranged final battle between all the existing national governments; the military units are very possibly robotic or entirely digital in nature, so as to avoid wasting materiel / human life.    They ultimately win the battle, just barely; I think the girl protagonist gives them an edge in the competition by hacking the timing / control system in favor of their team, causing a discrepancy which only the boy protagonist notices due to his heightened perceptions.  It was agreed that the winning government gets dominion over one of the few islands of land left that was not irradiated in some previous war; the protagonists get to live there as a reward for their victory.  I believe the boy protagonist later realizes he's gradually losing his ability to exactly quantify the passage of time.  Published no later than 2004, likely somewhere between 1980-2000, probably sometime in the '90s. Why do we need to discard one dummy variable?