Are there rings with uncountably many irreducible elements (prime elements, if in a PID)?

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
2
down vote

favorite












I'm playing around and trying to construct rings with different numbers of irreducible elements, hence the above question.






elementary-set-theory ring-theory







share|cite|improve this question









New contributor




Federica Zanni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.















  • 1




    Massive overkill: since being irreducible/prime is a first-order condition, take a nontrivial ultrapower of any ring with infinitely many irreducible elements.
    – Noah Schweber
    3 hours ago










  • @NoahSchweber: The term "first-order" is very common but seems to be one used in different branches of mathematics with different meanings. And ultrafilter is also unfamiliar to me. But if a a result in algebra can be proved from other branches I'd be happy to learn more about that branch. Is that logic?
    – P Vanchinathan
    2 hours ago














up vote
2
down vote

favorite












I'm playing around and trying to construct rings with different numbers of irreducible elements, hence the above question.






elementary-set-theory ring-theory







share|cite|improve this question









New contributor




Federica Zanni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.















  • 1




    Massive overkill: since being irreducible/prime is a first-order condition, take a nontrivial ultrapower of any ring with infinitely many irreducible elements.
    – Noah Schweber
    3 hours ago










  • @NoahSchweber: The term "first-order" is very common but seems to be one used in different branches of mathematics with different meanings. And ultrafilter is also unfamiliar to me. But if a a result in algebra can be proved from other branches I'd be happy to learn more about that branch. Is that logic?
    – P Vanchinathan
    2 hours ago












up vote
2
down vote

favorite









up vote
2
down vote

favorite











I'm playing around and trying to construct rings with different numbers of irreducible elements, hence the above question.






elementary-set-theory ring-theory







share|cite|improve this question









New contributor




Federica Zanni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











I'm playing around and trying to construct rings with different numbers of irreducible elements, hence the above question.






elementary-set-theory ring-theory




elementary-set-theory ring-theory






share|cite|improve this question









New contributor




Federica Zanni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




Federica Zanni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited 3 hours ago









Andrés E. Caicedo

64k7156239




64k7156239






New contributor




Federica Zanni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 3 hours ago









Federica Zanni

112




112




New contributor




Federica Zanni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





Federica Zanni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






Federica Zanni is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







  • 1




    Massive overkill: since being irreducible/prime is a first-order condition, take a nontrivial ultrapower of any ring with infinitely many irreducible elements.
    – Noah Schweber
    3 hours ago










  • @NoahSchweber: The term "first-order" is very common but seems to be one used in different branches of mathematics with different meanings. And ultrafilter is also unfamiliar to me. But if a a result in algebra can be proved from other branches I'd be happy to learn more about that branch. Is that logic?
    – P Vanchinathan
    2 hours ago












  • 1




    Massive overkill: since being irreducible/prime is a first-order condition, take a nontrivial ultrapower of any ring with infinitely many irreducible elements.
    – Noah Schweber
    3 hours ago










  • @NoahSchweber: The term "first-order" is very common but seems to be one used in different branches of mathematics with different meanings. And ultrafilter is also unfamiliar to me. But if a a result in algebra can be proved from other branches I'd be happy to learn more about that branch. Is that logic?
    – P Vanchinathan
    2 hours ago







1




1




Massive overkill: since being irreducible/prime is a first-order condition, take a nontrivial ultrapower of any ring with infinitely many irreducible elements.
– Noah Schweber
3 hours ago




Massive overkill: since being irreducible/prime is a first-order condition, take a nontrivial ultrapower of any ring with infinitely many irreducible elements.
– Noah Schweber
3 hours ago












@NoahSchweber: The term "first-order" is very common but seems to be one used in different branches of mathematics with different meanings. And ultrafilter is also unfamiliar to me. But if a a result in algebra can be proved from other branches I'd be happy to learn more about that branch. Is that logic?
– P Vanchinathan
2 hours ago




@NoahSchweber: The term "first-order" is very common but seems to be one used in different branches of mathematics with different meanings. And ultrafilter is also unfamiliar to me. But if a a result in algebra can be proved from other branches I'd be happy to learn more about that branch. Is that logic?
– P Vanchinathan
2 hours ago










1 Answer
1






active

oldest

votes

















up vote
5
down vote













Take a polynomial ring in a single variable $x$ with coefficients in real numbers (or complex numbers, what is important is the uncountablity). Then all degree one polynomials $(x-alpha)$ are irreducible. Being degree 1 they cannot be products of lower degree polynomials!
This being a PID irreducible, prime are one and the same.






share|cite|improve this answer




















    Your Answer




    StackExchange.ifUsing("editor", function ()
    return StackExchange.using("mathjaxEditing", function ()
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    );
    );
    , "mathjax-editing");

    StackExchange.ready(function()
    var channelOptions =
    tags: "".split(" "),
    id: "69"
    ;
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function()
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled)
    StackExchange.using("snippets", function()
    createEditor();
    );

    else
    createEditor();

    );

    function createEditor()
    StackExchange.prepareEditor(
    heartbeatType: 'answer',
    convertImagesToLinks: true,
    noModals: false,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    );



    );






    Federica Zanni is a new contributor. Be nice, and check out our Code of Conduct.









     

    draft saved


    draft discarded


















    StackExchange.ready(
    function ()
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2977040%2fare-there-rings-with-uncountably-many-irreducible-elements-prime-elements-if-i%23new-answer', 'question_page');

    );

    Post as a guest

















    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    5
    down vote













    Take a polynomial ring in a single variable $x$ with coefficients in real numbers (or complex numbers, what is important is the uncountablity). Then all degree one polynomials $(x-alpha)$ are irreducible. Being degree 1 they cannot be products of lower degree polynomials!
    This being a PID irreducible, prime are one and the same.






    share|cite|improve this answer
























      up vote
      5
      down vote













      Take a polynomial ring in a single variable $x$ with coefficients in real numbers (or complex numbers, what is important is the uncountablity). Then all degree one polynomials $(x-alpha)$ are irreducible. Being degree 1 they cannot be products of lower degree polynomials!
      This being a PID irreducible, prime are one and the same.






      share|cite|improve this answer






















        up vote
        5
        down vote










        up vote
        5
        down vote









        Take a polynomial ring in a single variable $x$ with coefficients in real numbers (or complex numbers, what is important is the uncountablity). Then all degree one polynomials $(x-alpha)$ are irreducible. Being degree 1 they cannot be products of lower degree polynomials!
        This being a PID irreducible, prime are one and the same.






        share|cite|improve this answer












        Take a polynomial ring in a single variable $x$ with coefficients in real numbers (or complex numbers, what is important is the uncountablity). Then all degree one polynomials $(x-alpha)$ are irreducible. Being degree 1 they cannot be products of lower degree polynomials!
        This being a PID irreducible, prime are one and the same.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 3 hours ago









        P Vanchinathan

        14.4k12036




        14.4k12036




















            Federica Zanni is a new contributor. Be nice, and check out our Code of Conduct.









             

            draft saved


            draft discarded


















            Federica Zanni is a new contributor. Be nice, and check out our Code of Conduct.












            Federica Zanni is a new contributor. Be nice, and check out our Code of Conduct.











            Federica Zanni is a new contributor. Be nice, and check out our Code of Conduct.













             


            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2977040%2fare-there-rings-with-uncountably-many-irreducible-elements-prime-elements-if-i%23new-answer', 'question_page');

            );

            Post as a guest
































































            -

            Popular posts from this blog

            How to check contact read email or not when send email to Individual?

            Image
            Clash Royale CLAN TAG #URR8PPP up vote 1 down vote favorite I'm using WordPress 4.9.8, CiviCRM to 5.5.1, I usually send email to contact by Search> Find contacts View contact details Action> Send email Send email ok, Contact received mail ok like picture But status only Email sent though contact read email or not. So, can CiviCRM can change status to Email read when contact read email? wordpress email share | improve this question asked Sep 26 at 0:12 ToanLuong 49 9 add a comment  |  up vote 1 down vote favorite I'm using WordPress 4.9.8, CiviCRM to 5.5.1, I usually send email to contact by Search> Find contacts View contact details Action> Send email Send email ok, Contact received mail ok like picture But status only Email sent though contact read email or not. So, can CiviCRM can change status to Email read when contact read email? wordpress email share | improve this questi...
            Read more

            How many registers does an x86_64 CPU actually have?

            Image
            Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite I am currently learning reverse engineering and am studying the flags register. I had in my mind that rflags was just another name for one of the 16 general purpose registers, for example rax or rbx . But it looks like rflags is actually an additional register. So that makes 17 registers in total... how many more could there be? I have spent at least an hour on this and found numerous different answers. The best answer so far is this, which says that there are 40 registers in total. 16 General Purpose Registers 2 Status Registers 6 Code Segment Registers 16 SSE Registers 8 FPU/MMX Registers But if I add that up, I get 48. Could anybody provide an official answer on how many registers an x86_64 CPU has (e.g. an Intel i7). Additionally, I have seen references to 'hardware' and 'architectural' registers. What are those registers and how many are there? register x86-64 share | improve this...
            Read more

            Displaying single band from multi-band raster using QGIS

            Image
            Clash Royale CLAN TAG #URR8PPP 1 How can I extract a single band from multi-band raster in QGIS? I have an remote sensed image which has 6 bands (including NDVI band), I want to display each band separately, but have no idea how to do. I have seen some questions similar here but none worked for me. The original image (has 6 bands) is: I want to display the band 6 which should be like this: But I tried gdal_translate, and couldn't get the correct result. What I have got is: qgis raster multi-band share | improve this question edited Mar 5 at 0:53 Summer asked Mar 4 at 6:42 Summer Summer 23 6 Is this any help gis.stackexchange.com/questions/220658/… ? if not gis.stackexchange.com/questions/62133/… might help. – Michael Stimson Mar 4 at 6:46 Thanks for answering but when I used gdal_translate, qgis showed that 'Error 4: Kayena.tif: No such file or directory". Would you know how to fi...
            Read more