Talk:Cancellable Semiring with Unity is Additive Semiring

Earlier on today I posted up the definition of the Trivial Ring (i.e. one in which the ring product is always zero.

I noticed that the underlying group structure need not be abelian for the ring axioms to be satisfied. This is contrary to the usual ring definition, which insists that it does need to be abelian.

This result, it appears, when I look at it more closely, does not hold for the trivial ring. That is, a structure consisting of a non-abelian group with a zero ring product obeys axioms A, M0, M1 and D of Ring but is not technically a ring as such.

I believe that if it's specified that the ring product is not universally zero, then this result holds, but I think I need to work on it a bit more.

Thoughts, anyone?