Let $M_1$ and $M_2$ be commutative monoids, $M_1$ written additively with identity $0$ and $M_2$ multiplicatively with identity $1$. Furthermore, let $M_2$ act on the left on $M_1$ via monoid endomorphisms, that is, we have a homomorphism of monoids $\varphi:M_2\rightarrow\mathrm{End}(M_1)$; as usual, instead of $\varphi(m_2)(m_1)$ for $m_1\in M_1,m_2\in M_2$, we write $m_2\cdot m_1$ or just $m_2m_1$. Then we can define a commutative monoid structure on the cartesian product $M_1\times M_2$ with identity $(0,1)$ and the binary operation $(m_1,m_2)\odot(m_1',m_2'):=(m_2'm_1+m_2m_1',m_2m_2')$. This is a bit like the semidirect product of monoids, but the action is applied to both first entries. If we choose, for instance, $\varphi$ to be the trivial homomorphism, then this is the direct product of $M_1$ and $M_2$.

My question simply is: Has this construction already been studied? And if so, what is known about it? Does it have a name? Thanks in advance!