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Let $R$ be a ring. $M,N$ be two submodules of $W$. Let

$$0 \longrightarrow M\longrightarrow M+N\longrightarrow H\longrightarrow 0$$ and $$0 \longrightarrow N\longrightarrow M+N\longrightarrow L\longrightarrow 0$$ be two split short exact sequences in ${\rm Mod}R$, where ${\rm Mod}R$ is the category of all right $R$-modules, and the maps $M\longrightarrow M+N$ and $N\longrightarrow M+N$ are two inclusion maps. Question:

Can we obtain the following two isomorphisms

$$M\cong （M\cap N） \oplus (M/(M\cap N)）$$ and $$N\cong （M\cap N） \oplus (N/(M\cap N)）$$?

Junling Zheng
  • 23

1 Answers1

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Not in general.

For example, take $R=k[x]$, the ring of polynomials over a field $k$, take $W=R/(x)\oplus R$, $M=0\oplus R$, and $N$ the submodule of $W$ generated by $(1,1)$, so both $M$ and $N$ are isomorphic as $R$-modules to $R$.

Then $W=M+N$, and if $X=R/(x)\oplus 0$ then $W=M\oplus X=N\oplus X$, so both inclusions $M\to W$ and $N\to W$ split.

But $M\cap N=0\oplus xR\cong R$ and $M/(M\cap N)\cong R/(x)$, so $M\not\cong (M\cap N)\oplus M/(M\cap N)$.

Jeremy Rickard
  • 28,899