Name/references for analogue of ring with 2-cocycle condition instead of distributivity - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T02:06:10Z http://mathoverflow.net/feeds/question/43247 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43247/name-references-for-analogue-of-ring-with-2-cocycle-condition-instead-of-distribu Name/references for analogue of ring with 2-cocycle condition instead of distributivity Vipul Naik 2010-10-22T23:21:02Z 2010-10-22T23:21:02Z <p>I'm looking for a name for, and any past study on, the following kind of algebraic structure:</p> <p>A set <em>S</em> 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.:</p> <p>$$(x * (y + z)) + (y * z) = ((x + y) * z) + (x * y)$$</p> <p>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.</p>