Ideal generated in a noetherian ring

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I read a proof of a theorem but there is a part that I don't quite understand: If $A$ is a Noetherian Ring and if $J$ is an ideal of $A$ of finite type generated by $(a_i)_iin Iin A^I$, then there exist $i_1, i_2,ldots, i_nin I$ such that J is generated by $a_i_1, a_i_2,ldots,a_i_n$



Thanks in advance.



P.S. I know that any Ideal of a Noetherian Ring is of finite type.




ring-theory




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    I read a proof of a theorem but there is a part that I don't quite understand: If $A$ is a Noetherian Ring and if $J$ is an ideal of $A$ of finite type generated by $(a_i)_iin Iin A^I$, then there exist $i_1, i_2,ldots, i_nin I$ such that J is generated by $a_i_1, a_i_2,ldots,a_i_n$



    Thanks in advance.



    P.S. I know that any Ideal of a Noetherian Ring is of finite type.




    ring-theory




    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
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      down vote

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      I read a proof of a theorem but there is a part that I don't quite understand: If $A$ is a Noetherian Ring and if $J$ is an ideal of $A$ of finite type generated by $(a_i)_iin Iin A^I$, then there exist $i_1, i_2,ldots, i_nin I$ such that J is generated by $a_i_1, a_i_2,ldots,a_i_n$



      Thanks in advance.



      P.S. I know that any Ideal of a Noetherian Ring is of finite type.




      ring-theory




      share|cite|improve this question













      I read a proof of a theorem but there is a part that I don't quite understand: If $A$ is a Noetherian Ring and if $J$ is an ideal of $A$ of finite type generated by $(a_i)_iin Iin A^I$, then there exist $i_1, i_2,ldots, i_nin I$ such that J is generated by $a_i_1, a_i_2,ldots,a_i_n$



      Thanks in advance.



      P.S. I know that any Ideal of a Noetherian Ring is of finite type.





      ring-theory





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      edited 23 hours ago









      dmtri

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          Let $(b_1,dots, b_r)$ be a finite set of generators for $J$. Each of these generators is a finite linear combination of elements in $(a_i)_iin I$. For each $jenspace(1le jle r)$, denote $S_j$ the set of the $i$s in $I$ involved in an expression of $b_j$ as a linear combination of elements in $(a_i)_iin I$. Then the set
          $$G=bigla_imid iin S_1cupdotscup S_rbigr$$
          is a finite set of generators for $J$, contained in the initial set $(a_i)_iin I$.






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            Let $(b_1,dots, b_r)$ be a finite set of generators for $J$. Each of these generators is a finite linear combination of elements in $(a_i)_iin I$. For each $jenspace(1le jle r)$, denote $S_j$ the set of the $i$s in $I$ involved in an expression of $b_j$ as a linear combination of elements in $(a_i)_iin I$. Then the set
            $$G=bigla_imid iin S_1cupdotscup S_rbigr$$
            is a finite set of generators for $J$, contained in the initial set $(a_i)_iin I$.






            share|cite|improve this answer

























              up vote
              1
              down vote



              accepted










              Let $(b_1,dots, b_r)$ be a finite set of generators for $J$. Each of these generators is a finite linear combination of elements in $(a_i)_iin I$. For each $jenspace(1le jle r)$, denote $S_j$ the set of the $i$s in $I$ involved in an expression of $b_j$ as a linear combination of elements in $(a_i)_iin I$. Then the set
              $$G=bigla_imid iin S_1cupdotscup S_rbigr$$
              is a finite set of generators for $J$, contained in the initial set $(a_i)_iin I$.






              share|cite|improve this answer























                up vote
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                accepted







                up vote
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                down vote



                accepted






                Let $(b_1,dots, b_r)$ be a finite set of generators for $J$. Each of these generators is a finite linear combination of elements in $(a_i)_iin I$. For each $jenspace(1le jle r)$, denote $S_j$ the set of the $i$s in $I$ involved in an expression of $b_j$ as a linear combination of elements in $(a_i)_iin I$. Then the set
                $$G=bigla_imid iin S_1cupdotscup S_rbigr$$
                is a finite set of generators for $J$, contained in the initial set $(a_i)_iin I$.






                share|cite|improve this answer













                Let $(b_1,dots, b_r)$ be a finite set of generators for $J$. Each of these generators is a finite linear combination of elements in $(a_i)_iin I$. For each $jenspace(1le jle r)$, denote $S_j$ the set of the $i$s in $I$ involved in an expression of $b_j$ as a linear combination of elements in $(a_i)_iin I$. Then the set
                $$G=bigla_imid iin S_1cupdotscup S_rbigr$$
                is a finite set of generators for $J$, contained in the initial set $(a_i)_iin I$.







                share|cite|improve this answer













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                share|cite|improve this answer











                answered 23 hours ago









                Bernard

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