nonuniform local ring with essential socle











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I need an example: Is there any local non uniform ring $R$ with essential socle and its socle is exactly direct sum of ALL minimal left ideals? Thanks.






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    I need an example: Is there any local non uniform ring $R$ with essential socle and its socle is exactly direct sum of ALL minimal left ideals? Thanks.






    abstract-algebra ring-theory modules







    share|cite|improve this question
























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I need an example: Is there any local non uniform ring $R$ with essential socle and its socle is exactly direct sum of ALL minimal left ideals? Thanks.






      abstract-algebra ring-theory modules







      share|cite|improve this question













      I need an example: Is there any local non uniform ring $R$ with essential socle and its socle is exactly direct sum of ALL minimal left ideals? Thanks.






      abstract-algebra ring-theory modules




      abstract-algebra ring-theory modules






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      asked 2 days ago









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          There is no such example.



          That is, let $R$ be a local ring with essential left socle which is the direct sum of all minimal left ideals.



          Since $R$ is local, all the minimal left ideals of $R$ are isomorphic to $R/M$ where $M$ is the maximal ideal.



          Suppose $phi :L_1to L_2$ is an isomorphism of two such minimal left ideals. Then necessarily $L_1oplus L_2$ is contained in the socle. But ${x+phi(x)mid xin L_1}$ is another left ideal of $R$ which is easily seen to be isomorphic to $L_1$ and $L_2$. Including this copy of $L_1$ in the sum would make it fail to be a direct sum.



          So, the left socle is necessarily just one minimal left ideal. Since you have also assumed it is essential, it is in fact contained in every other left ideal, making the ring left uniform.






          share|cite|improve this answer





















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            There is no such example.



            That is, let $R$ be a local ring with essential left socle which is the direct sum of all minimal left ideals.



            Since $R$ is local, all the minimal left ideals of $R$ are isomorphic to $R/M$ where $M$ is the maximal ideal.



            Suppose $phi :L_1to L_2$ is an isomorphism of two such minimal left ideals. Then necessarily $L_1oplus L_2$ is contained in the socle. But ${x+phi(x)mid xin L_1}$ is another left ideal of $R$ which is easily seen to be isomorphic to $L_1$ and $L_2$. Including this copy of $L_1$ in the sum would make it fail to be a direct sum.



            So, the left socle is necessarily just one minimal left ideal. Since you have also assumed it is essential, it is in fact contained in every other left ideal, making the ring left uniform.






            share|cite|improve this answer

























              up vote
              0
              down vote













              There is no such example.



              That is, let $R$ be a local ring with essential left socle which is the direct sum of all minimal left ideals.



              Since $R$ is local, all the minimal left ideals of $R$ are isomorphic to $R/M$ where $M$ is the maximal ideal.



              Suppose $phi :L_1to L_2$ is an isomorphism of two such minimal left ideals. Then necessarily $L_1oplus L_2$ is contained in the socle. But ${x+phi(x)mid xin L_1}$ is another left ideal of $R$ which is easily seen to be isomorphic to $L_1$ and $L_2$. Including this copy of $L_1$ in the sum would make it fail to be a direct sum.



              So, the left socle is necessarily just one minimal left ideal. Since you have also assumed it is essential, it is in fact contained in every other left ideal, making the ring left uniform.






              share|cite|improve this answer























                up vote
                0
                down vote










                up vote
                0
                down vote









                There is no such example.



                That is, let $R$ be a local ring with essential left socle which is the direct sum of all minimal left ideals.



                Since $R$ is local, all the minimal left ideals of $R$ are isomorphic to $R/M$ where $M$ is the maximal ideal.



                Suppose $phi :L_1to L_2$ is an isomorphism of two such minimal left ideals. Then necessarily $L_1oplus L_2$ is contained in the socle. But ${x+phi(x)mid xin L_1}$ is another left ideal of $R$ which is easily seen to be isomorphic to $L_1$ and $L_2$. Including this copy of $L_1$ in the sum would make it fail to be a direct sum.



                So, the left socle is necessarily just one minimal left ideal. Since you have also assumed it is essential, it is in fact contained in every other left ideal, making the ring left uniform.






                share|cite|improve this answer












                There is no such example.



                That is, let $R$ be a local ring with essential left socle which is the direct sum of all minimal left ideals.



                Since $R$ is local, all the minimal left ideals of $R$ are isomorphic to $R/M$ where $M$ is the maximal ideal.



                Suppose $phi :L_1to L_2$ is an isomorphism of two such minimal left ideals. Then necessarily $L_1oplus L_2$ is contained in the socle. But ${x+phi(x)mid xin L_1}$ is another left ideal of $R$ which is easily seen to be isomorphic to $L_1$ and $L_2$. Including this copy of $L_1$ in the sum would make it fail to be a direct sum.



                So, the left socle is necessarily just one minimal left ideal. Since you have also assumed it is essential, it is in fact contained in every other left ideal, making the ring left uniform.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 2 days ago









                rschwieb

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