Where $I$ being nontrivial means $I \neq 0$ and $I \neq R$. I have found that such a ring must not be Noetherian (since otherwise, roughly, you could take some nonzero nonunit $x_1 \in I$ and recursively take $x_i \in I$ such that $x_i^2 = x_{i-1}$ to create a strictly increasing chain of ideals), and if we don't require our ring to be an integral domain, the ring $\mathbb{Z}/6\mathbb{Z}$ with ideal $I = (2)$ works.
If $I$ is a non-zero proper ideal of a commutative integral domain, is it possible for $I$ to be its own square?
Is casting the same thing as converting? In Jesse Liberty's Learning C# book, he says "Objects of one type can be converted into objects of another type. This is called casting."  If you investigate the IL generated from the code below, you can clearly see that the casted assignment isn't doing the same thing as the converted assignment. In the former, you can see the boxing/unboxing occurring; in the latter you can see a call to a convert method.   I know in the end it may be just a silly  semantic difference--but is casting just another word for converting. I don't mean to be snarky, but I'm not interested in anyone's gut feeling on this--opinions don't count here! Can anyone point to a definitive reference that confirms or denies if casting and converting are the same thing?      object x;     int y;      x = 4;      y = ( int )x;      y = Convert.ToInt32( x );   Thank you  rp  Note added after Matt's comment about explicit/implicit:  I don't think implicit/explicit is the difference. In the code I posted, the change is explicit in both cases. An implicit conversion is what occurs when you assign a short to an int.  Note to Sklivvz:  I wanted confirmation that my suspicion of the looseness of Jesse Liberty's (otherwise usually lucid and clear) language was correct. I thought that Jesse Liberty was being a little loose with his language. I understand that casting is routed in object hierarchy--i.e., you can't cast from an integer to a string but you could cast from custom exception derived from System.Exception to a System.Exception.   It's interesting, though, that when you do try to cast from an int to a string the compiler tells you that it couldn't "convert" the value. Maybe Jesse is more correct than I thought! How do you set the expiry date or a maximum age in the HTTP headers for static resources in IIS I am trying to think of a non-colloquial word or phrase that means "practice what you preach." I would appreciate any suggestions. Thank you! Compiling LaTeX document empties .bbl file I'm trying out biblatex for the document I'm compiling, using the biber backend on my miktex distribution. Everything, including biber, should be up to date, assuming the MiKTeX repository currently holds the latest versions of everything.  This is what the biblatex documentation (p. 101) says about how often I have to run biber to update my .bbl file:     Whenever a reference to a work which has not been cited before is   added, this procedure must be repeated. This is also the case if the   last reference to a work which has been cited before is removed   because some citation labels may change in this case.   So if I make changes to my document (which is a very common activity in the process of writing a document), but leave the citations unchanged, I should not have to re-run biber before compiling the changes. However, every time I compile my document, regardless of the compiler or documentclass, the .bbl file is emptied. Consequently, the next time I compile the document without running biber, the bibliography is removed (seeing as the .bbl file contains nothing), and all my citations are broken. For this reason, I have to run biber before every time I compile the document, which largely counters the benefit of externalizing the bibliography in the first place.  How can I make the compilers stop overwriting my .bbl file and make everything behave as described in the biblatex manual?  Upon investigating the contents of the auxiliary files, I found that the .bbl file (after running biber, before compiling) begins with the following lines:  % $ biblatex auxiliary file $ % $ biblatex bbl format version 2.3 $   Does the biblatex version 2.8 still mark the .bbl files as version 2.3? If not, could this be the problem? And if so, why does my 2.8 version of biblatex (partly) behave as version 2.3?  Any help solving this problem would be much appreciated. I'm writing my final report in LaTeX for my bachelor's degree. I'm aiming at having a document similar to the .doc template used normally.   I want to have my customs headers and footers applied on pages when I'm starting a new chapter.  I'm using documentclass report with fancyhdr and lastpage packages.  My headers are defined like this:  \lhead{\myTitle} \rhead{\today} \cfoot{\thepage\ of \pageref{LastPage}}   My chapters are created using standard \chapter{title} macro.  On pages with new chapters, the only headers/footers I see is the standard page number at the bottom.  What should I do?  EDIT: I've finally gone with Peter Breitfeld's solution. I'Ve gone a step further and created my own chapter command:  \newcommand{\myChapter}[1]{     \chapter{#1}     \thispagestyle{fancy} }   Thanks for the help! What to do with bounty on question which is no longer relevant? What does "cpu MHz" field mean in the /proc/cpuinfo of a hyper-threading cpu? The CPU is a i7-3770@3.40GHz. It has 4 cores and each core has 2 threads. Here is the dmidecode output:  # dmidecode -t 4 # dmidecode 2.9 SMBIOS 2.7 present.  Handle 0x0042, DMI type 4, 42 bytes Processor Information     Socket Designation: SOCKET 0     Type: Central Processor     Family: &lt;OUT OF SPEC&gt;     Manufacturer: Intel(R) Corporation     ID: A9 06 03 00 FF FB EB BF     Version: Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz     Voltage: 1.1 V     External Clock: 100 MHz     Max Speed: 3800 MHz     Current Speed: 3400 MHz     Status: Populated, Enabled     Upgrade: &lt;OUT OF SPEC&gt;     L1 Cache Handle: 0x003F     L2 Cache Handle: 0x003E     L3 Cache Handle: 0x0040     Serial Number: Not Specified     Asset Tag: Fill By OEM     Part Number: Fill By OEM     Core Count: 4     Core Enabled: 4     Thread Count: 8     Characteristics:         64-bit capable   It will be 8 logic core in a system, like what shows in /proc/cpuinfo. But can any one tell why the cpu MHz of a core is 1600MHz? I guess there is 2 threads in a core, so a hw thread freq may be about the half of the core's? How this number is calculated?  processor   : 7 vendor_id   : GenuineIntel cpu family  : 6 model       : 58 model name  : Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz stepping    : 9 cpu MHz     : 1600.000 cache size  : 8192 KB physical id : 0 siblings    : 8 core id     : 3 cpu cores   : 4 apicid      : 7 initial apicid  : 7 fpu     : yes fpu_exception   : yes cpuid level : 13 wp      : yes flags       : fpu vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx rdtscp lm constant_tsc arch_perfmon pebs bts rep_good xtopology nonstop_tsc aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx smx est tm2 ssse3 cx16 xtpr pdcm sse4_1 sse4_2 x2apic popcnt aes xsave avx lahf_lm ida arat tpr_shadow vnmi flexpriority ept vpid bogomips    : 7013.49 clflush size    : 64 cache_alignment : 64 address sizes   : 36 bits physical, 48 bits virtual power management:   Also, here is the output of lshw and lscpu command. There are also 1600MHz mentioned.   lshw info:  #lshw -class processor   *-cpu                           description: CPU        product: Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz        vendor: Intel Corp.        physical id: 42        bus info: cpu@0        version: Intel(R) Core(TM) i7-3770 CPU @ 3.40GHz        slot: SOCKET 0        size: 1600MHz        capacity: 3800MHz        width: 64 bits        clock: 100MHz        capabilities: fpu fpu_exception wp vme de pse tsc msr pae mce cx8 apic sep mtrr pge mca cmov pat pse36 clflush dts acpi mmx fxsr sse sse2 ss ht tm pbe syscall nx rdtscp x86-64 constant_tsc arch_perfmon pebs bts rep_good xtopology nonstop_tsc aperfmperf pni pclmulqdq dtes64 monitor ds_cpl vmx smx est tm2 ssse3 cx16 xtpr pdcm sse4_1 sse4_2 x2apic popcnt aes xsave avx lahf_lm ida arat tpr_shadow vnmi flexpriority ept vpid cpufreq   lscpu info:  #lscpu Architecture:          x86_64 CPU op-mode(s):        32-bit, 64-bit CPU(s):                8 Thread(s) per core:    2 Core(s) per socket:    4 CPU socket(s):         1 NUMA node(s):          1 Vendor ID:             GenuineIntel CPU family:            6 Model:                 58 Stepping:              9 CPU MHz:               1600.000 Virtualization:        VT-x L1d cache:             32K L1i cache:             32K L2 cache:              256K L3 cache:              8192K I'm trying to make a conditional formatting rule in Google Spreadsheet that would highlight a row green based on text in multiple cells.   If cells L2, M2, and N2 = APPROVED; and cell O2 = X; then row 2 = highlighted green. Add Item to Finder Sidebar I tried to create a Windows 10 USB boot medium with WoeUSB. I formatted a USB drive with an NTFS partition, but WoeUSB complains with:  Installation failed! Exit code: 256 Log: WoeUSB v@@WOEUSB_VERSION@@ ============================== Mounting source filesystem... Error: File "/media/woeusb_source_1543626298_6098/sources/install.wim" in source image has exceed the FAT32 Filesystem 4GiB Single File Size Limitation and cannot be installed.  You must specify a different --target-filesystem. Refer: https://github.com/slacka/WoeUSB/wiki/Limitations#fat32-filesystem-4gib-single-file-size-limitation for more info. Unmounting and removing "/media/woeusb_source_1543626298_6098"... You may now safely detach the target device   I've also tried to start WoeUSB via command line, did not work.  sudo woeusb --partition Win10_1809Oct_English_x64.iso /dev/sdb   My iso is located in:     home/sawyer/Downloads/Win10_1809Oct_English_x64.iso  Thanks in advance! I've been trying to get windows installed for three nights now, so any help towards that goal is greatly appreciated. How can I make a macOS Mojave bootable USB installer in Windows 10? Sounds of the letter a How can I know, precisely, when to differentiate the sounds of the letter a, like in: apple and vault? Transparent path/stroke fill in Illustrator Remove array element based on object property What is the difference between [@] and [*] when referencing bash array values? says:  If the index number is @ or *, all members of an array are referenced.  When I do this: LIST=(1 2 3) for i in &quot;${LIST[@]}&quot;; do   echo &quot;example.$i&quot; done  it gives the desired result: example.1 example.2 example.3  But when I use ${LIST[*]}, I get example.1 2 3  instead. Why? Edit: when using printf, @ and * actually do give the same results. While learning about Random variables I came across the mean of random variable X.  The definition says that the expected value of random variable E(X) = Mean of Random variable X  I am not able to understand why is that so. Can any one please help me with it SemanticImport vs Import differ on same CSV file? Types in MySQL: BigInt(20) vs Int(20) A Monk threw three coloured darts, then spends one minute searching the battlefield, how many darts does the Monk recover? I dug a large room to specifications, 3 blocks tall and 16 blocks wide as well as long. I dug it at 10 levels above bedrock, in a location {(-240, 0) -> (-224, 16)} specified by  slime finder app. I am definitely more than 24 blocks away and less than 32 away from the spawn room, as specified by the wiki .  Yet, no slimes. I have been sitting in my location patiently listening for the moist slapping sounds of slime. I have gone in to the room every time I heard a phantom slap but still nothing.  Am I just being dumb somehow or do the minecraft lords just hate me?  If it helps/anyone cares here's my seed: 6405460532872068998