A Ring structure on the Class of Combinatorial Games

J. Conway defined useful operations on the Class of combinatorial games and also introduced a notion of equivalence between games. Conway showed that, under his equivalence, games form a Group. However, Conway product is not well defined on equivalence classes of arbitrary games (though it is well defined for surreals). We consider an equivalence relation finer than Conway’s and show that under such a relation combinatorial games actually form a Ring. We hint to other possible relations on the Class of combinatorial games.