I know that if (x)={0} then the if 0=r0 such that r belongs to R therefor it's a field. Most likely I'm wrong but I need help with the second part if the ideal is R
Let $R$ be a commutative ring with identity. Show that $R$ is a field if and only if the only ideals of $R$ are $R$ itself and the zero ideal $(0)$.  I can't figure out where to start other that I need to prove some biconditional statement. Any help?
A single word for someone who spends their wealth very foolishly $\mathbb{R}P^4$ and $\mathbb{R}P^6$ do not admit fields of tangent $2$-planes See the related question . This is the second part of question 4-C in Milnor and Stasheff's book on characteristic classes. In the solution to the first part, we rely on the fact that having a field of tangent $1$-planes allows us to split the tangent bundle into a direct sum. It doesn't seem like this should be true for a field of tangent $2$-planes (otherwise the question wouldn't just be asking about $P^4$ and $P^6$), so I'm not sure exactly where to start.  edit: actually, since $P^n$ can be given a Riemannian metric, any subbundle should be a Whitney summand; this doesn't answer the question as to why they only ask for $P^4$ and $P^6$ though, so I'm still confused. Sending signals to USB port using C I want to send a simple 5V signal to the usb port. The USB cabel's insulation will be removed and connected to a relay which is connected to an electric door system. I think I should write this code in C. Can somebody give me a hint? I was wanting to initialize some strings at the top of my script with variables that have no yet been set, such as:  str1='I went to ${PLACE} and saw ${EVENT}' str2='If you do ${ACTION} you will ${RESULT}'   and then later on PLACE, EVENT, ACTION, and RESULT will be set. I want to then be able to print my strings out with the variables expanded. Is my only option eval? This seems to work:  eval "echo ${str1}"   is this standard? is there a better way to do this? It would be nice to not run eval considering the variables could be anything. Can the manual and tome magic items that increase stats be used multiple times by bypassing the century wait time via the spell Sequester? Assume we have a Wizard who is at least level 13 and can reliably cast  many times (either through acquired spell scrolls, or has acquired enough material components for it to be a non-issue), has a safe place (through the spell ) to be under the influence of Sequester for many centuries, and has access to the Manual of Bodily Health, Manual of Gainful Exercise, Manual of Quickness of Action, Tome of Clear Thought, Tome of Leadership and Influence, and Tome of Understanding.  Per the item descriptions in the DMG:     ...your [STAT] score increases by 2, as does your maximum for that score. The manual then loses its magic, but regains it in a century.   Now, normally this would be a once-in-a-lifetime use, or maybe twice (due to the lifespan of the races), but if a wizard used Sequester on themself, many years can pass without growing older:     ...[willing target creature] falls into a state of suspended animation. Time ceases to flow for it, and it doesn't grow older. You can set a condition for the spell to end [before the spell is dispelled by the caster].   So, assuming that time continues to flow for the magic items, but not for the sequestered wizard, can they study these books and take the necessary long rests, Sequester for a hundred years,  “wake” from the Sequester, and then repeat an arbitrary-but-finite number of times for an arbitrarily high (but finite) improvement to their stats?  To end the cycle and escape the demiplane, the wizard would simply Plane Shift out:     ...You can specify a target destination in general terms ... and you appear in or near that destination. How do I run a sudo command in Emacs? Odd behavior in a QGIS plugin: my function is triggered twice I have a plugin with a function (triggered by a button) that selects a specific layer by name, then shows its attribute table. When I close the plugin and open it again, the attribute table pops up twice, and repeating this keeps adding another table popup. What's wrong?  Code:  def showTable(self):     for lyrs in QgsMapLayerRegistry.instance().mapLayers().values():         if lyrs.name() == "TestFeatures":             lyrTest = lyrs     self.iface.showAttributeTable(lyrTest) Do the -ing and to-infinitive "verbs" that follow catenative verbs always take the grammatical function of "noun"? I'm wondering whether or not the verb form that follows a catenative verb has the grammatical function of a noun or of a verb, and whether or not it depends on the first catenative verb.  "I like to run" "I like running"  the -ing and the to infinitive can easily be switched in this sentence, and it seems to be implying "I like [the act of[running]]", so both the infinitive and the gerund are nouns.  "I like to run marathons"/"I like running marathons"   These seem like they might still be nouns but need "verb powers" to take the object, and then these sentences really turn into "I like [the act of [running marathons]]"/ "I like [to be in the state of [running marathons]]".  Is this true for both the -ing (and so it would be a gerund) and the to-infinitive? Are they, sort of, initially verbs so that they can take the object, and then the verb + object becomes a noun phrase?  If this is true, I'm also wondering if the same rules apply for a catenative verb such as "start".  "I started to run in the mornings" "I started running marathons in 2008"  "start" is a catenative verb that isn't a cognitive/emotion verb, and specifically suggests action. So what is the grammatical function of the to-infinitive and -ing verb forms? are they verbs? are they nouns? It just seems incorrect to say that their grammatical function is a noun, but everything I've seen on the internet is pretty vague. I've also seen that to-infinitives are never verbs, but only from one source and it just doesn't seem right to me.  So any ideas on the true grammatical function of the -ing and the to-infinitive after a catenative verb? And is it universal across catenative verbs, or does it depend on the verb category? Is a memory leak created if a MemoryStream in .NET is not closed? I have the following code:  MemoryStream foo(){     MemoryStream ms = new MemoryStream();     // write stuff to ms     return ms; }  void bar(){     MemoryStream ms2 = foo();     // do stuff with ms2     return; }   Is there any chance that the MemoryStream that I've allocated will somehow fail to be disposed of later?    I've got a peer review insisting that I manually close this, and I can't find the information to tell if he has a valid point or not. Implement ```-style (fenced) Markdown code blocks Currently, Stack Exchange’s Markdown parser only allows four-space indents to represent code blocks:      // some code     // another line of code    and other Markdown implementations allow for an alternative syntax, that doesn’t require indenting each code line:  ``` // some code // another line of code ```   This is much more convenient to type out. It would be super useful if Stack Exchange could support this syntax.    By extension, this syntax also allow you to specify the source language right after the opening ```:  ```js // some code // another line of code ```   …which would then enable syntax highlighting for that specific language. Although it’s interesting metadata, I don’t think this feature is needed on Stack Overflow, as the syntax highlighting library works pretty well for various languages.  So, even if you would allow this syntax but ignore the ```language, this would greatly improve my productivity on Stack Exchange.  What do you think?  The moderators are currently collecting feedback regarding this feature, so please post a comment with your thoughts. Good idea? Bad idea? Don’t really care? What do you think are the benefits/drawbacks? Experiences? Let them know! Thanks! Every time I call \section{**Section Name**}, a number appears to the left of the name of the section.  How do I hide this number? Choice of ECC curve on USB token I would like to get a USB token to secure my keys. My use case is protection of 3 GnuPG keys that I will be using 10 times per day at least. I plan to create a new key ring from scratch. Because ECC seems more future-oriented than RSA, this is what I chose to use. I'm wondering which USB token to choose as well as which curve.  On  it is said that many people think curves recommended by the NIST or defined by Brainpool have a doubtful origin; therefore, they recommend the non-standardized Bernstein’s Curve 25519. On  one poster says that    He is not aware of any profound criticism of Brainpool curves, and Bernstein’s Curve 25519 is hard to protect against side-channel attacks when implemented in embedded devices.   As a result, I'm a bit lost as to which key/curve to choose. Has the concept of non-integer $(n+m)$-dimensional spacetime ever been investigated by theoretical physicists? The following image serves to aid the reader in understanding the "privileged character" of $3+1$-spacetime.    The wikipedia article on , and the  "The priveleged character of $3+1$-dimensional spacetime" in particular, made me think a bit about the possibility that we might live in a non-integer amount of spatial and/or time dimensions.   The notion of attaching a non-negative real number to a metric space has at least mathematically already been described by such concepts as "" and "".   This may sound silly/ignorant/absurd to professional practicing theoretical physicists. To me (a layman), however, it doesn't sound much stranger than the idea of wrapping up six extra dimensions (which is, from what I understand, considered to be a serious possibility by those who study $10$-dimensional String Theory) into intricate shapes called "Calabi-Yau Manifolds".   Has any research on $(a +b)$-dimensional spacetime (where $a,b \in \mathbb{R}_{\geq 0} $) ever been done? If so, what where the findings? If not, why not? There is a restriction on setting your birthday:     Oops! There was a problem updating your profile:   - Birthday must be after 1920/01/01   I realize the number of 90 year old users is small, but shouldn't the restriction be a bit more relaxed, such as larger than any ?  Either 120 or 150 seems appropriate.  For example, my grandfather turns 90 next year (and just barely misses the current cutoff), but I can't imagine he's the only one around that age who is computer literate and still active enough to be interested in (the increasingly broad range of) SE sites.  I don't think he has contributed yet (mostly because of the comment restriction, I'd see him commenting before asking or answering), but I have emailed him with links that I know he's read.  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