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Let $M$ be $R$-module and $K,K',L,L'$ be submodules of $M$ with the property that $M=K\oplus K'=L\oplus L'$. Prove that if $K=L$ then $K'\cong L'$. Furthermore, if $H$ is submodule of $M$ and $K$ is submodule of $H$ then $H=K\oplus\left(H\cap K'\right)$

Thanks for any insight.

user110834
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First thing: suppose $K\oplus K'=K\oplus L'$ then $K'\cong (K\oplus K')/K=(K\oplus L')/K\cong L$.

Second thing: by distributivity you get $H=(K+ K')\cap H=(K\cap H)+ (K'\cap H)=K+(K'\cap H)$. Notice that $K\cap (K'\cap H)\subset K\cap K'=0$ so $H=K\oplus(K'\cap H)$.

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user52045
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  • +1 vote :) I want to know, if we only suppose $K\cong L$, can conclude $K'\cong L'$? – Truong Nov 26 '13 at 16:39
  • No this wont be true. Its important in the proof that I m dividing by the same module not isomorphic ones. – user52045 Nov 26 '13 at 16:41
  • Can we have a counterexample? – user110834 Nov 26 '13 at 16:43
  • For example there are some noncommutative rings with the property that $R=R\oplus R$ as modules but ofc we can write $0\oplus R=R\oplus R$ and it would mean that $R\cong 0$ with is silly. – user52045 Nov 26 '13 at 16:45
  • Example of such ring simpliest as far as i know http://math.stackexchange.com/questions/570217/isomorphism-of-r-modules – user52045 Nov 26 '13 at 16:48
  • Much simpler example $\bigoplus_\mathbb Z\mathbb Z\oplus 0\cong \bigoplus_\mathbb Z\mathbb Z\oplus \bigoplus_\mathbb Z\mathbb Z$ – user52045 Nov 26 '13 at 17:36
  • Ehm... If $K\cap L=0$, can conclude that $K+L$ is direct summand of $M$? – Truong Nov 27 '13 at 08:59