For modules, let $M = M_1 ⊕ M_2$ and let $f :M→N$ be an epimorphism with $K = \ker f$ and $N = f(M_1) + f (M_2)$.
(1) Prove that if $K= ( K \cap M_1)+ (K \cap M_2)$, then this sum is direct.
Could someone give me hints about this question. I am having troubles about how to solve this question. Thanks a lot.
Let $x_1 , x_2$ be elements of $M_1$ and $M_2$ respectively. Assume that $f(x_1) + f(x_2) = 0$. We want to show that $f(x_1) = f(x_2) = 0$.
We know that $x_1 + x_2 \in K$. So, by the hypothesis, there exist $y_1, y_2$ in $(K\cap M_1)$ and $(K\cap M_2)$ respectively such that $x_1+x_2 = y_1+y_2$. Now, we notice that $x_1-y_1= y_2-x_2 \in M_1 \cap M_2 = \lbrace 0 \rbrace$.
Thus $x_1 = y_1$ and $x_2 = y_2$. Finally, since $y_1, y_2 \in K$, we get $f(x_1) = f(x_2) = 0$, as desired.