How to prove that if A, B, C are submodules of M module then $ A \cap (B+C) = (A\cap B)+C \iff C \subseteq A $
Any help would be appreciated!
How to prove that if A, B, C are submodules of M module then $ A \cap (B+C) = (A\cap B)+C \iff C \subseteq A $
Any help would be appreciated!
Suppose that $A\cap(B+C)=(A\cap B)+C$; since $C\subseteq (A\cap B)+C=A\cap(B+C)\subseteq A$, it follows that $C\subseteq A$.
Conversely, suppose that $C\subseteq A$. If $x\in A\cap (B+C)$, then $x\in A$ and there exist $b\in B$, $c\in C$ such that $x=b+c$. Then $b\in B$, and $b=x-c\in A+C = A$ (since $C\subseteq A$), so $b\in (A\cap B)$. Therefore, $x = b+c\in (A\cap B)+C$. Thus, $A\cap(B+C)\subseteq (A\cap B)+C$.
Conversely, if $x\in (A\cap B)+C$, then there exists $y\in A\cap B$ and $c\in C$ such that $x=y+c$. Then $x\in A$, since $y\in A$, $c\in C\subseteq A$, so $y+c\in A$. And $y\in B$, $c\in C$, so $x=y+c\in B+C$. Therefore, $x\in A\cap(B+C)$. This proves $(A\cap B)+C\subseteq A\cap(B+C)$, proving the equality.