A derivation $D$ is a map on an algebra $A$ over a field $F$, $D:A \rightarrow A$ such that $D$ is an $F$-linear map and the following holds:
$$D(ab)=aD(b)+D(a)b$$ $$\forall ab\in A$$
Is there a name for the similar construct where $D$ is a map with $A$ an algebra with the field $F$, $D:A \rightarrow F$ such that again:
$$D(a+b) = D(a) + D(b)$$ $$D(ab) = D(b)a + D(a)b$$ $$\forall a,b\in A$$
Such a map goes by the name of derivation.