Commutative algebra for the Conway games I was reading the book On Number And Games and I have some question. In this book Conway constructed the set of "games" with a addition and a multiplication. I understand that the surreal numbers are a particular case of such games. My question is if it games form a commutative ring. And if it is true, does the ring games form a integral domain? has finite Krull dimension? Can one describe its maximal spectrum?. And finally, is there any book about these questions?.
 A: See the correction below.
The class of games does indeed constitute a commutative ring under Conway's definitions of sums and products. This is already in Conway's book, but more explicitly stated and proved in Combinatorial Game Theory
by Aaron N. Siegel, American Mathematical Society, 2013. On the other hand, the theorem that ensures that the cancellation law holds for multiplication is explicitly formulated and proved for surreal numbers (Theorem 8, pp.19-20 for Conway; Proposition 2.3, p.413 for Siegel). I suspect it fails for games.
Correction (1/13/22): (Siegel p. 413) establishes the following for the multiplication of long games. For all long games $x,y,z$: (a) $x0 \cong 0$, (b) $xy \cong yx$, (c) $x1 \cong x$, (d) $(xy)z=x(yz)$, (e) $(x+y)z=xz +yz$. Siegel's long games are Conway's games.
However, as David E Speyer observes in his comments, contrary to what I previously claimed this is not enough to show that the class of games constitutes a commutative ring under Conway's definitions of sums and products. See Speyer's comments for details.
