Let $R$ be a commutative ring, $M=M_1\bigoplus M_2$ be a left $R$-module and $f:M_1\to M_1,m_1\mapsto rm_1$ be a homothety of $M_1$, then the map $h(m)=(rm_1,m_2)$ is a module endomorphism of $M$.
My question: Is $h$ a homothety of $M$? Is it possible to consider $h$ as $h(m)=(f(m_1),m_2)=(rm_1,m_2)=r(m_1,m_2)=rm$? I know it is not a restriction to $M_1$.