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Let $R$ be a unital ring, and let $M$ be an $R$-module. Suppose $M_1, \ldots, M_n$ are $R$-submodules of $M$ such that $M_i + M_j=M$, for $i \neq j$.

Are the modules $M/(\cap_{i=1}^n M_i)$ and $\bigoplus_{i=1}^n (M/M_i)$ isomorphic? And what if $R$ is a finite dimensional algebra over a field?

There is a natural homomorphism of modules from $M$ to $\bigoplus_{i=1}^n (M/M_i)$, which has kernel $\cap_{i=1}^n M_i$. In the case $n=2$, I can prove that this homomorphism is surjective (hence, the modules mentioned in the previous paragraph are isomorphic). But I do not know what to do for $n \geq 3$, and also, I cannot find a couter-example.

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This is typically not true for $n>2$. For instance, suppose $R=k$ is a field, $M=k^2$, and the $M_i$ are a bunch of distinct $1$-dimensional subspaces. Then the $M_i$ satisfy your hypotheses, but $M/\bigcap M_i=M$ is $2$-dimensional while $\bigoplus M/M_i$ is $n$-dimensional.

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