Let $A$ and $B$ be unital associative algebras over some field $k$. I can form the cartesian product $A\times B$ with multiplication $(a,b)(a',b')=(aa',bb')$ to get a new associated unital algebra.
But what I can also do is considering a (unital) algebra-homomorphism $$\varphi:A\to\text{End} (B)$$ and consider the algebra with underlying set $A\times B$ but now with the multiplication $$(a,b)(a',b'):=(aa',b\varphi(a)(b')).$$
This is again a $k$-algebra but with a different multiplication. The first case is a special case for $\varphi(a)=\text{id}$ for all $a\in A$.
Does this structure has a name?