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.

  • $\begingroup$ Do you know that distributivity does not follow from the cocycle condition? $\endgroup$ – user5810 Oct 22 '10 at 23:34
  • $\begingroup$ Yes, I have plenty of examples in the context of a specific application; in fact practically all the interesting examples of cocycles aren't bilinear (i.e., distributive). $\endgroup$ – Vipul Naik Oct 23 '10 at 3:34

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