Let us consider the following short exact sequence $$0 \to M_1 \oplus M_2 \stackrel{f\oplus id}{\to} N \oplus M_2 \to P \to 0$$ such that $Ext^1(P, M_1) =0.$ Then I am wondering for a section $P \to N \oplus M_2.$
My Arguments are as follows: Since $Ext^1(P, M_1) =0.$, Therefore there is a section $s : P \to N$ for the short exact sequence $$0 \to M_1 \stackrel{f}{\to} N \to P \to 0$$. Then the map $i_N \circ s$ will serve as a section the first short exact sequence. Where, $ i_N : N \to N \oplus M_2$ is the map given by $i_N(n) =(n, 0)$
Is the argument correct?
thank you in advance. Any help will be appreciated.