Finite Integral Extension of DVRs $rk_R(A)$












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Let $R subset A$ be a finite integral extension of discrete valuation rings such that $A = oplus_{i=1} ^n R$ therefore $rk_R(A)=n$. Denote by $K_R,K_A$ the quotient fields of $R$ and $A$. Obviously this gives a rise for field extension $K_A / K_R$ with $d := [K_A: K_R]$.



My goal is to verify that $n =d$.



My ideas: $K_R$ arises from $R$ as localisation on the multiplicative set $S:= R backslash {0}$. Therefore $K_R= RS^{-1}$. Since localisations comute with direct sums we obtain



$$AS^{-1} = oplus_{i=1} ^n K_R$$



If we could deduce that $AS^{-1}= K_A$ we are done but I'm not sure if it true.



Is there maybe a more effective way to show the claim?










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    $begingroup$


    Let $R subset A$ be a finite integral extension of discrete valuation rings such that $A = oplus_{i=1} ^n R$ therefore $rk_R(A)=n$. Denote by $K_R,K_A$ the quotient fields of $R$ and $A$. Obviously this gives a rise for field extension $K_A / K_R$ with $d := [K_A: K_R]$.



    My goal is to verify that $n =d$.



    My ideas: $K_R$ arises from $R$ as localisation on the multiplicative set $S:= R backslash {0}$. Therefore $K_R= RS^{-1}$. Since localisations comute with direct sums we obtain



    $$AS^{-1} = oplus_{i=1} ^n K_R$$



    If we could deduce that $AS^{-1}= K_A$ we are done but I'm not sure if it true.



    Is there maybe a more effective way to show the claim?










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Let $R subset A$ be a finite integral extension of discrete valuation rings such that $A = oplus_{i=1} ^n R$ therefore $rk_R(A)=n$. Denote by $K_R,K_A$ the quotient fields of $R$ and $A$. Obviously this gives a rise for field extension $K_A / K_R$ with $d := [K_A: K_R]$.



      My goal is to verify that $n =d$.



      My ideas: $K_R$ arises from $R$ as localisation on the multiplicative set $S:= R backslash {0}$. Therefore $K_R= RS^{-1}$. Since localisations comute with direct sums we obtain



      $$AS^{-1} = oplus_{i=1} ^n K_R$$



      If we could deduce that $AS^{-1}= K_A$ we are done but I'm not sure if it true.



      Is there maybe a more effective way to show the claim?










      share|cite|improve this question









      $endgroup$




      Let $R subset A$ be a finite integral extension of discrete valuation rings such that $A = oplus_{i=1} ^n R$ therefore $rk_R(A)=n$. Denote by $K_R,K_A$ the quotient fields of $R$ and $A$. Obviously this gives a rise for field extension $K_A / K_R$ with $d := [K_A: K_R]$.



      My goal is to verify that $n =d$.



      My ideas: $K_R$ arises from $R$ as localisation on the multiplicative set $S:= R backslash {0}$. Therefore $K_R= RS^{-1}$. Since localisations comute with direct sums we obtain



      $$AS^{-1} = oplus_{i=1} ^n K_R$$



      If we could deduce that $AS^{-1}= K_A$ we are done but I'm not sure if it true.



      Is there maybe a more effective way to show the claim?







      abstract-algebra ring-theory commutative-algebra extension-field






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      asked Dec 30 '18 at 0:23









      KarlPeterKarlPeter

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