Finite Length Modules











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Let $R$ be a ring. If $M, N$ and $L$ are $R$-modules, with, $ell(M), ell(N), ell(L) < infty,$ and $M times N cong M times L,$ it is true that $N cong L?$



The same occurs replacing $times$ by $oplus$ or $otimes?$



When the converse is true? That is, if $M times N cong M times L, $$N cong L$ and (additional hypothesis), then $ell(M), ell(N), ell(L) < infty$?



For example, we know that, if $ell(M), ell(N) < infty,$ then $ell(M otimes N) le ell(M) ell(N),$ but this is not useful in a general case.






ring-theory modules tensor-products direct-sum







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    Let $R$ be a ring. If $M, N$ and $L$ are $R$-modules, with, $ell(M), ell(N), ell(L) < infty,$ and $M times N cong M times L,$ it is true that $N cong L?$



    The same occurs replacing $times$ by $oplus$ or $otimes?$



    When the converse is true? That is, if $M times N cong M times L, $$N cong L$ and (additional hypothesis), then $ell(M), ell(N), ell(L) < infty$?



    For example, we know that, if $ell(M), ell(N) < infty,$ then $ell(M otimes N) le ell(M) ell(N),$ but this is not useful in a general case.






    ring-theory modules tensor-products direct-sum







    share|cite|improve this question
























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      Let $R$ be a ring. If $M, N$ and $L$ are $R$-modules, with, $ell(M), ell(N), ell(L) < infty,$ and $M times N cong M times L,$ it is true that $N cong L?$



      The same occurs replacing $times$ by $oplus$ or $otimes?$



      When the converse is true? That is, if $M times N cong M times L, $$N cong L$ and (additional hypothesis), then $ell(M), ell(N), ell(L) < infty$?



      For example, we know that, if $ell(M), ell(N) < infty,$ then $ell(M otimes N) le ell(M) ell(N),$ but this is not useful in a general case.






      ring-theory modules tensor-products direct-sum







      share|cite|improve this question













      Let $R$ be a ring. If $M, N$ and $L$ are $R$-modules, with, $ell(M), ell(N), ell(L) < infty,$ and $M times N cong M times L,$ it is true that $N cong L?$



      The same occurs replacing $times$ by $oplus$ or $otimes?$



      When the converse is true? That is, if $M times N cong M times L, $$N cong L$ and (additional hypothesis), then $ell(M), ell(N), ell(L) < infty$?



      For example, we know that, if $ell(M), ell(N) < infty,$ then $ell(M otimes N) le ell(M) ell(N),$ but this is not useful in a general case.






      ring-theory modules tensor-products direct-sum




      ring-theory modules tensor-products direct-sum






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      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










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