Am Donnerstag 08 Oktober 2009 03:05:13 schrieb Felipe Lessa:
On Wed, Oct 07, 2009 at 08:44:27PM -0400, Jason McCarty wrote:
Daniel Fischer wrote:
Am Mittwoch 07 Oktober 2009 23:51:54 schrieb Joe Fredette:
I generally find semirings defined as a ring structure without additive inverse and with 0-annihilation (which one has to assume in the case of SRs, I included it in my previous definition because I wasn't sure if I could prove it via the axioms, I think it's possible, but I don't recall the proof).
0*x = (0+0)*x = 0*x + 0*x ==> 0*x = 0
This proof only works if your additive monoid is cancellative, which need not be true in a semiring. The natural numbers extended with infinity is one example (if you don't take 0*x = 0 as an axiom, I think there are two possibilities for 0*∞).
It was a proof for a ring (with or without unit), which Joe stated above he didn't recall. There your additive monoid is cancellative since it's a group :D
Given that
x = 1*x = (0+1)*x = 0*x + 1*x = 0*x + x
we can show that
x = x + 0*x (right) x = 0*x + x (left)
so, by definition of 'zero', we have that 0*x is a zero.
Not necessarily, we don't know 0*x + y = y for arbitrary y yet. If the additive monoid isn't cancellative, that needn't be the case. In Jason's example, you can indeed set 0*∞ = ∞.
But we can easily prove that there can be only one zero: suppose we have two zeros z1 and z2; it follows that
z1 = z1 + z2 = z2
So 0*x = 0. Any flaws?
-- Felipe.