Let $A$ be a commutative ring with only finitely many minimal prime ideals.   Is the zero ideal $(0)$ decomposable?  [The converse implication is well known. Recall that an ideal is decomposable if it is a finite intersection of primary ideals.]
Recently I encountered a problem (the first exercise from chapter four of Atiyah &amp; McDonald's Introduction to Commutative Algebra) stating that if $\mathfrak{a}$ is a decomposable ideal of $A$ (a commutative ring with unity), then the prime spectrum of $A/ \mathfrak{a}$ has finitely many irreducible components. This follows easily from the recognition that the maximal irreducible subspaces of $\textrm{ Spec } (A / \mathfrak{a})$ are precisely the "zero loci" of the minimal prime ideals of $A /\mathfrak{a}$.   I'm curious about the converse - the proof isn't easily reversed since the notion of minimal ideals of $\mathfrak{a} $ doesn't make sense before we know what we are trying to prove. My intuition says that it is false based on the general premise that images are badly-behaved (and the fact that it isn't part of the exercise.) However, I've had some difficulty constructing a counterexample, so that the main purpose of this post is to ask for a reasonable procedure or heuristic for doing so (or, of course, proof that my intuition is false.)   If it helps, if $\textrm{Spec }(A / \mathfrak{a})$ is irreducible then the nilradical $\mathcal{R}_{A /\mathfrak{a}}$ is prime so that $r(\mathfrak{a}) = \rho^{-1} ( \mathcal{R}_{A / \mathfrak{a}} ) = \displaystyle\cap_{i=1}^n \rho^{-1} (p_i),$ where $p_i$ are the minimal prime ideals of $A /\mathfrak{a}$ and $\rho $ is the associated projection, is also prime. Thanks!
Series with denominator that is square of product of consecutive numbers Background I was ignorant as an undergraduate and didn't focus heavily in any specific area. I finally graduated with a single math/computer science degree that consisted of 5 math classes beyond multivariable calculus and a few relatively basic computer science classes (algorithm analysis, SQL database design, etc.). My only research experience is in economics. Now, I work at a research institution in economics where I primarily write Stata/MATLAB code. Question I'd like to get more mathematics under my belt over the next few years in order to either enroll in a PhD program in economics/finance or a M.S. program in finance. What are my options for getting more math experience that I can use in graduate school admissions? Options These are the options I thought of so far, but I would like to know more if possible.  Study enough courses through a system like Open CourseWare to gain enough experience to enroll in more advanced undergraduate classes part time, which my employer might pay for, and then ideally move into an M.S. program in applied math.  Return to my alma mater for a M.S. degree in math that would at least give me a basis to either move into another math graduate program with higher rankings or a lower-ranked graduate program in the fields of my choice.  Join the Air Force and hopefully take advantage of the GI bill to take more courses somewhere and gain entrance into an M.S. program in applied math.  (Sadly) Forgo advanced mathematics and find a consulting job, which although unpleasant in the field of economics (in my opinion), wouldn't be difficult to obtain given my research background in the field.   Do I have any other options? I also plan to take the GRE subject test in mathematics to prove that I have at least minimal knowledge, regardless of what course I pursue (since many graduate programs will require it). Why is cold fusion considered bogus? Cold fusion is being mentioned a lot lately because of some new setup that apparently works. This is an unverified claim.  See for example:         ( of that last link in the Wayback Machine, given frequent  from that page.)   While we should give the scientific community time to evaluate the set up and eventually replicate the results, there is undoubtedly some skepticism that cold fusion would work at all, because the claim is quite extraordinary.  In the past, after Fleischmann and Pons announced their cold fusion results, in perfectly good faith, they were proven wrong by subsequent experiments.  What are the experimental realities that make Fleischmann and Pons style cold fusions experiments easy to get wrong?  Would the same risks apply to this new set up? I know that there are questions asking like "intersection of a infinite collection of sets" and I can understand that the answer for that one is a null set, but I got a question here, in which all sets are finite and nonempty. Please take a look at the pic below.  It's a True/False question.  What I'm confused about is that the "infinity" sign in the interception. Since all sets are finite, does it mean that there are many sets in the chain are the same? (because the symbol used is for subsets not for proper subsets). Please give me some hints how I should think about this question.  Thank you How to make a different font and color of text in labels? Alignment issue in Area 51 main site - site switcher, inbox and achievements Making dual power supply from single for opamp? For the circuit of opamp 741. we have to supply the dual power for it. How can I make a dual power supply from single. For example, make a (+9V)-GND-(-9V) from single 9V battery ? Norm of adjoint operator in Hilbert space Wrapped hexagon topology When a square is wrapped at the edges, obtains a torus. Is it possible to do the same for a hexagon?  What is the shape/topology of wrapping a hexagon?     Note: I was just reading this  and came up with this question. I'm a physicist. Why couldn't the Jedi detect Senator Palpatine as having powers? In Return of the Jedi, when trying to sneak into the base on a shuttle, Luke said he senses Darth Vader's presence and said he shouldn't have come lest he jeopardize the mission. Darth Vader also sensed Luke, but let him in anyways. Also, Yoda and Obi-Wan could detect the Force in young Anakin.  So if Jedi and Sith can detect each other, why couldn't they detect Senator Palpatine to begin with? It seems like a giant plot hole. Get most common value for each value of another column in SQL I have a table like this:   Column  | Type | Modifiers  ---------+------+-----------  country | text |   food_id | int  |   eaten   | date |    And for each country, I want to get the food that is eaten most often.  The best I can think of (I'm using postgres) is:  CREATE TEMP TABLE counts AS     SELECT country, food_id, count(*) as count FROM munch GROUP BY country, food_id;  CREATE TEMP TABLE max_counts AS     SELECT country, max(count) as max_count FROM counts GROUP BY country;  SELECT country, max(food_id) FROM counts     WHERE (country, count) IN (SELECT * from max_counts) GROUP BY country;   In that last statement, the GROUP BY and max() are needed to break ties, where two different foods have the same count.  This seems like a lot of work for something conceptually simple.  Is there a more straight forward way to do it? Expanding GCDs Reasoning: $\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$ I'm just wondering the reason behind being able to do this:  $$\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$$  Is there an elementary proof to the reasoning behind this? In the search results of stackexchange.com/search?q=*, allow people to click on a tag and go to that website with the tag page open.   For example,  shows me this:        But clicking on the hovered-on tag takes me to  and the panda.   Instead, it should take me to    The math.SE URL is already there in  /#question-summary-608993 &gt; div.summary &gt; div.started.fr &gt; a #question-summary-608993 &gt; div.hot-question-site-icon &gt; a Optional Android notifications for reputation changes I would like to get a notification if my reputation count has changed. I think there is already some internal API for that. (I can see that changes in the menu bar in the top).  I'm sure that some people would not like that so that should be optional. Why must $a$ and $b$ both be coprime when proving that the square root of two is irrational? Suppose we wish to prove that the square root of two is irrational. We begin by assuming that it is rational.  Namely, where both $a$ and $b$ are integers $$\frac{a}{b} = \sqrt 2  % MathType!MTEF!2!1!+- % faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0x % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa % aiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaamOyaaaacq % GH9aqpdaGcaaqaaiaaikdaaSqabaaaaa!31D3! $$  Why is it so important that both $a$ and $b$ be coprime and the fraction be irreducible?  That is, if $a$ and $b$ are not coprime and the fraction is indeed reducible, then$$\frac{a}{b} \ne \sqrt 2  % MathType!MTEF!2!1!+- % faaagCart1ev2aaaKnaaaaWenf2ys9wBH5garuavP1wzZbqedmvETj % 2BSbqefm0B1jxALjharqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC0x % bbL8FesqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yqaq % pepae9pg0FirpepeKkFr0xfr-xfr-xb9Gqpi0dc9adbaqaaeGaciGa % aiaabeqaamaabaabaaGcbaWaaSaaaeaacaWGHbaabaGaamOyaaaacq % GHGjsUdaGcaaqaaiaaikdaaSqabaaaaa!3294! $$ What do *args and **kwargs mean? What exactly do *args and **kwargs mean?  According to the Python documentation, from what it seems, it passes in a tuple of arguments.  def foo(hello, *args):     print hello      for each in args:         print each  if __name__ == '__main__':     foo("LOVE", ["lol", "lololol"])   This prints out:  LOVE ['lol', 'lololol']   How do you effectively use them? Bibtex true unsort Using Google Music Manager in Linux from the command line My goal is to use Google Music Manager on a Linux server.  I'm wondering if the music manager can operate from the command line and with no GUI.  If I could set the software up to watch a particular directory on the server and upload from there, that would be the only necessary functionality.  Thank you. Citing @inbook with biblatex: how to put authors/editors before book title? I am not an expert of random forest but I clearly understand that the key issue with random forest is the (random) tree generation. Can you explain me how the trees are generated? (i.e. What is the used distribution for tree generation?)  Thanks in advance !