induced maximal ideals in ring of functions [closed]












1












$begingroup$


Let $R$ be a ring and $X$ be a non-empty set.

We know that $operatorname{Map}(X,R) = {f : X to R}$ is a ring

where, $(f+g)(x)= f(x) + g(x)$ and $(fcdot g)(x)=f(x)cdot g(x)$


I want to know that if $M$ is a maximal ideal in $R$ whether this ideal induces a maximal ideal in $operatorname{Map}(X,R)$?






abstract-algebra







share|cite|improve this question











$endgroup$



closed as off-topic by amWhy, user26857, Saad, KReiser, mrtaurho Dec 27 '18 at 12:37


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – amWhy, user26857, Saad, KReiser, mrtaurho

If this question can be reworded to fit the rules in the help center, please edit the question.





















    1












    $begingroup$


    Let $R$ be a ring and $X$ be a non-empty set.

    We know that $operatorname{Map}(X,R) = {f : X to R}$ is a ring

    where, $(f+g)(x)= f(x) + g(x)$ and $(fcdot g)(x)=f(x)cdot g(x)$


    I want to know that if $M$ is a maximal ideal in $R$ whether this ideal induces a maximal ideal in $operatorname{Map}(X,R)$?






    abstract-algebra







    share|cite|improve this question











    $endgroup$



    closed as off-topic by amWhy, user26857, Saad, KReiser, mrtaurho Dec 27 '18 at 12:37


    This question appears to be off-topic. The users who voted to close gave this specific reason:


    • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – amWhy, user26857, Saad, KReiser, mrtaurho

    If this question can be reworded to fit the rules in the help center, please edit the question.



















      1












      1








      1





      $begingroup$


      Let $R$ be a ring and $X$ be a non-empty set.

      We know that $operatorname{Map}(X,R) = {f : X to R}$ is a ring

      where, $(f+g)(x)= f(x) + g(x)$ and $(fcdot g)(x)=f(x)cdot g(x)$


      I want to know that if $M$ is a maximal ideal in $R$ whether this ideal induces a maximal ideal in $operatorname{Map}(X,R)$?






      abstract-algebra







      share|cite|improve this question











      $endgroup$




      Let $R$ be a ring and $X$ be a non-empty set.

      We know that $operatorname{Map}(X,R) = {f : X to R}$ is a ring

      where, $(f+g)(x)= f(x) + g(x)$ and $(fcdot g)(x)=f(x)cdot g(x)$


      I want to know that if $M$ is a maximal ideal in $R$ whether this ideal induces a maximal ideal in $operatorname{Map}(X,R)$?






      abstract-algebra




      abstract-algebra






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 26 '18 at 21:05









      user26857

      39.3k124183




      39.3k124183










      asked Dec 26 '18 at 14:27









      S. WalkerS. Walker

      92




      92




      closed as off-topic by amWhy, user26857, Saad, KReiser, mrtaurho Dec 27 '18 at 12:37


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – amWhy, user26857, Saad, KReiser, mrtaurho

      If this question can be reworded to fit the rules in the help center, please edit the question.







      closed as off-topic by amWhy, user26857, Saad, KReiser, mrtaurho Dec 27 '18 at 12:37


      This question appears to be off-topic. The users who voted to close gave this specific reason:


      • "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – amWhy, user26857, Saad, KReiser, mrtaurho

      If this question can be reworded to fit the rules in the help center, please edit the question.






















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          Not only one, but sometimes many. For each $xin X$ and maximal ideal $M$, you can consider the subset of functions $f$ such that $f(x)in M$. If you check, you'll find this is a maximal ideal of the set of functions.



          It is not always the case that these are all the maximal ideals, though.






          share|cite|improve this answer











          $endgroup$




















            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            1












            $begingroup$

            Not only one, but sometimes many. For each $xin X$ and maximal ideal $M$, you can consider the subset of functions $f$ such that $f(x)in M$. If you check, you'll find this is a maximal ideal of the set of functions.



            It is not always the case that these are all the maximal ideals, though.






            share|cite|improve this answer











            $endgroup$


















              1












              $begingroup$

              Not only one, but sometimes many. For each $xin X$ and maximal ideal $M$, you can consider the subset of functions $f$ such that $f(x)in M$. If you check, you'll find this is a maximal ideal of the set of functions.



              It is not always the case that these are all the maximal ideals, though.






              share|cite|improve this answer











              $endgroup$
















                1












                1








                1





                $begingroup$

                Not only one, but sometimes many. For each $xin X$ and maximal ideal $M$, you can consider the subset of functions $f$ such that $f(x)in M$. If you check, you'll find this is a maximal ideal of the set of functions.



                It is not always the case that these are all the maximal ideals, though.






                share|cite|improve this answer











                $endgroup$



                Not only one, but sometimes many. For each $xin X$ and maximal ideal $M$, you can consider the subset of functions $f$ such that $f(x)in M$. If you check, you'll find this is a maximal ideal of the set of functions.



                It is not always the case that these are all the maximal ideals, though.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited Dec 26 '18 at 21:05









                user26857

                39.3k124183




                39.3k124183










                answered Dec 26 '18 at 15:14









                rschwiebrschwieb

                107k12102251




                107k12102251















                    -

                    Popular posts from this blog

                    Ellipse (mathématiques)

                    Image
                    Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
                    Read more

                    Quarter-circle Tiles

                    Image
                    up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
                    Read more

                    Mont Emei

                    Image
                    Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
                    Read more