Localization of a direct productLocalization of finite direct product of ring at an elementIs every local ring the localization of some other ring?Localization at a prime and direct limitsPrime ideals in an arbitrary direct product of ringsInfinite direct product of C-M ringsIdempotent direct summands of ringsConditions for a ring to be a direct product of local ringsDirect Summands of PI Rings as Right IdealsDirect product of soclesLocalization of finite direct product of ring at an elementLocalization at prime idea of the infinite direct product of $Z/2Z$

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Localization of a direct product


Localization of finite direct product of ring at an elementIs every local ring the localization of some other ring?Localization at a prime and direct limitsPrime ideals in an arbitrary direct product of ringsInfinite direct product of C-M ringsIdempotent direct summands of ringsConditions for a ring to be a direct product of local ringsDirect Summands of PI Rings as Right IdealsDirect product of soclesLocalization of finite direct product of ring at an elementLocalization at prime idea of the infinite direct product of $Z/2Z$













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Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.










share|cite|improve this question









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    0












    $begingroup$


    Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.










    share|cite|improve this question









    $endgroup$














      0












      0








      0


      0



      $begingroup$


      Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.










      share|cite|improve this question









      $endgroup$




      Is the localization of a direct product of two rings at a maximal (or prime) ideal identified with a localization of one of them? I would appreciate for any detailed answer.







      abstract-algebra ring-theory commutative-algebra noncommutative-algebra






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      asked Jun 24 '14 at 7:59









      karparvarkarparvar

      2,7001716




      2,7001716




















          1 Answer
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          4












          $begingroup$

          First prove the following: if $S_isubset R_i$ is a multiplicative set, then $$(S_1times S_2)^-1(R_1times R_2)simeq S_1^-1R_1times S_2^-1R_2.$$



          A maximal ideal of $R_1times R_2$ is of the form $M_1times R_2$ or $R_1times M_2$ with $M_i$ maximal in $R_i$. To localize $R_1times R_2$ at $M_1times R_2$, consider the multiplicative set $$(R_1times R_2)setminus (M_1times R_2)=(R_1setminus M_1)times R_2,$$ use the previous isomorphism and recall that $S^-1R=0$ if $0in S$.






          share|cite|improve this answer











          $endgroup$








          • 1




            $begingroup$
            can you give more detail why the isomorphism is true?
            $endgroup$
            – annimal
            Dec 9 '14 at 15:26










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          $begingroup$

          First prove the following: if $S_isubset R_i$ is a multiplicative set, then $$(S_1times S_2)^-1(R_1times R_2)simeq S_1^-1R_1times S_2^-1R_2.$$



          A maximal ideal of $R_1times R_2$ is of the form $M_1times R_2$ or $R_1times M_2$ with $M_i$ maximal in $R_i$. To localize $R_1times R_2$ at $M_1times R_2$, consider the multiplicative set $$(R_1times R_2)setminus (M_1times R_2)=(R_1setminus M_1)times R_2,$$ use the previous isomorphism and recall that $S^-1R=0$ if $0in S$.






          share|cite|improve this answer











          $endgroup$








          • 1




            $begingroup$
            can you give more detail why the isomorphism is true?
            $endgroup$
            – annimal
            Dec 9 '14 at 15:26















          4












          $begingroup$

          First prove the following: if $S_isubset R_i$ is a multiplicative set, then $$(S_1times S_2)^-1(R_1times R_2)simeq S_1^-1R_1times S_2^-1R_2.$$



          A maximal ideal of $R_1times R_2$ is of the form $M_1times R_2$ or $R_1times M_2$ with $M_i$ maximal in $R_i$. To localize $R_1times R_2$ at $M_1times R_2$, consider the multiplicative set $$(R_1times R_2)setminus (M_1times R_2)=(R_1setminus M_1)times R_2,$$ use the previous isomorphism and recall that $S^-1R=0$ if $0in S$.






          share|cite|improve this answer











          $endgroup$








          • 1




            $begingroup$
            can you give more detail why the isomorphism is true?
            $endgroup$
            – annimal
            Dec 9 '14 at 15:26













          4












          4








          4





          $begingroup$

          First prove the following: if $S_isubset R_i$ is a multiplicative set, then $$(S_1times S_2)^-1(R_1times R_2)simeq S_1^-1R_1times S_2^-1R_2.$$



          A maximal ideal of $R_1times R_2$ is of the form $M_1times R_2$ or $R_1times M_2$ with $M_i$ maximal in $R_i$. To localize $R_1times R_2$ at $M_1times R_2$, consider the multiplicative set $$(R_1times R_2)setminus (M_1times R_2)=(R_1setminus M_1)times R_2,$$ use the previous isomorphism and recall that $S^-1R=0$ if $0in S$.






          share|cite|improve this answer











          $endgroup$



          First prove the following: if $S_isubset R_i$ is a multiplicative set, then $$(S_1times S_2)^-1(R_1times R_2)simeq S_1^-1R_1times S_2^-1R_2.$$



          A maximal ideal of $R_1times R_2$ is of the form $M_1times R_2$ or $R_1times M_2$ with $M_i$ maximal in $R_i$. To localize $R_1times R_2$ at $M_1times R_2$, consider the multiplicative set $$(R_1times R_2)setminus (M_1times R_2)=(R_1setminus M_1)times R_2,$$ use the previous isomorphism and recall that $S^-1R=0$ if $0in S$.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Mar 13 at 6:40

























          answered Jun 24 '14 at 9:46









          user26857user26857

          39.4k124183




          39.4k124183







          • 1




            $begingroup$
            can you give more detail why the isomorphism is true?
            $endgroup$
            – annimal
            Dec 9 '14 at 15:26












          • 1




            $begingroup$
            can you give more detail why the isomorphism is true?
            $endgroup$
            – annimal
            Dec 9 '14 at 15:26







          1




          1




          $begingroup$
          can you give more detail why the isomorphism is true?
          $endgroup$
          – annimal
          Dec 9 '14 at 15:26




          $begingroup$
          can you give more detail why the isomorphism is true?
          $endgroup$
          – annimal
          Dec 9 '14 at 15:26

















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