What module is of finite length whose submodules aren't? [closed]











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Does such an example exist? It seems strange that a module whose submodules have infinite length has somehow infinite length






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closed as off-topic by Zvi, Lord Shark the Unknown, Brahadeesh, KReiser, Shailesh Nov 24 at 11:28


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    Does such an example exist? It seems strange that a module whose submodules have infinite length has somehow infinite length






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    closed as off-topic by Zvi, Lord Shark the Unknown, Brahadeesh, KReiser, Shailesh Nov 24 at 11:28


    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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Zvi, Brahadeesh, KReiser, Shailesh

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      Does such an example exist? It seems strange that a module whose submodules have infinite length has somehow infinite length






      modules







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      Does such an example exist? It seems strange that a module whose submodules have infinite length has somehow infinite length






      modules




      modules






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      asked Nov 22 at 16:47









      Ben-ZT

      1889




      1889




      closed as off-topic by Zvi, Lord Shark the Unknown, Brahadeesh, KReiser, Shailesh Nov 24 at 11:28


      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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Zvi, Brahadeesh, KReiser, Shailesh

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




      closed as off-topic by Zvi, Lord Shark the Unknown, Brahadeesh, KReiser, Shailesh Nov 24 at 11:28


      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 improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Zvi, Brahadeesh, KReiser, Shailesh

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          Let $R$ be a ring. An $R$-module $M$ has finite length if and only if it is artinian and noetherian. Since these two properties are inherited by $R$-submodules, if $Nleq M$ is an $R$-submodule of a finite-length module $M$, $N$ is artinian and noetherian. Hence it has finite length.






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            Let $R$ be a ring. An $R$-module $M$ has finite length if and only if it is artinian and noetherian. Since these two properties are inherited by $R$-submodules, if $Nleq M$ is an $R$-submodule of a finite-length module $M$, $N$ is artinian and noetherian. Hence it has finite length.






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              Let $R$ be a ring. An $R$-module $M$ has finite length if and only if it is artinian and noetherian. Since these two properties are inherited by $R$-submodules, if $Nleq M$ is an $R$-submodule of a finite-length module $M$, $N$ is artinian and noetherian. Hence it has finite length.






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                up vote
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                Let $R$ be a ring. An $R$-module $M$ has finite length if and only if it is artinian and noetherian. Since these two properties are inherited by $R$-submodules, if $Nleq M$ is an $R$-submodule of a finite-length module $M$, $N$ is artinian and noetherian. Hence it has finite length.






                share|cite|improve this answer












                Let $R$ be a ring. An $R$-module $M$ has finite length if and only if it is artinian and noetherian. Since these two properties are inherited by $R$-submodules, if $Nleq M$ is an $R$-submodule of a finite-length module $M$, $N$ is artinian and noetherian. Hence it has finite length.







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                answered Nov 23 at 7:18









                Blumer

                438110




                438110















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