Talk:Ring Product with Zero

This could be lifted to a more general result: In any ringoid where the distributand is a cancellative semigroup an element is a left or right zero of the distributor magma iff it is an additive identity (both left and right) of the distributand. Hence additive identity is both a left and right zero ie a zero of the distributor and distributor has an identity iff it has a left or right identity.

Basically the same proof showing distributor left or right identity iff distributand idempotent.

Separate (obvious) proof that any idempotent of cancellative semigroup is the identity.

"Ring Product with Zero" is a corollary.

Allows more "natural" development from positive natural numbers with addition to also have multiplication and then positive rationals long before conception of "zero" and later "negative".

Add either or both zero or identity to any semigroup, with simultaneous addition of additive identity and multiplicative zero for both positive naturals and rationals (and reals). --Arthur 12:53, 25 June 2011 (CDT)