Suppose that $M_1$, $M_2$, $N_1$, $N_2$ are $R$-module and $M_1\cong N_1$, $M_2\cong N_2$. Prove that $M_1\oplus M_2\cong N_1\oplus N_2$.
I know that
$M_1\cong N_1$ imply there exist an isomorphism $\phi_1:M_1\to N_1$.
$M_2\cong N_2$ imply there exist an isomorphism $\phi_2:M_2\to N_2$.
To prove $M_1\oplus M_2\cong N_1\oplus N_2$, it should be prove there exist an isomorphism $\phi:M_1\oplus M_2\to N_1\oplus N_2$.
I confuse to find there exist an isomorphism $\phi:M_1\oplus M_2\to N_1\oplus N_2$. Is it true that $M_1\oplus M_2=M_1+M_2$? and what the hint to prove this problem?
Best Answer
Just define: $$\phi:M_1 \oplus M_2 \to N_1 \oplus N_2$$ $$\phi(m_1,m_2)=(\phi_1(m_1),\phi_2(m_2))$$ And use their properties.