I'm looking for a name for, and any past study on, the following kind of algebraic structure:
A set S equipped with an additive operation making it an abelian group, and a multiplication $*:S \times S \to S$ satisfying the 2-cocycle condition over addition, i.e.:
$$(x * (y + z)) + (y * z) = ((x + y) * z) + (x * y)$$
for all $x,y,z \in S$. Of course, if we have both left and right distributivity, the cocycle condition follows, so any (not necessarily associative, commutative, or unital) ring gives an example of such a structure. Also, the 2-cocycle condition, along with either of the distributivity laws, implies the other.