Am Mittwoch 07 Oktober 2009 23:51:54 schrieb Joe Fredette:
I was just quoting from Hungerford's Undergraduate text, but yes, the "default ring" is in {Rng, Ring}, I haven't heard semirings used in the sense of a Rng.
It's been looong ago, I seem to have misremembered :? But there used to be a german term for Rngs, and it was neither Pseudoring nor quasiring, so I thought it was Halbring.
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
Wikipedia seems to agree with your definition, though it does have a note which says some authors use the definition of Abelian Group + Semigroup (my definition) as opposed to Abelian Group + Monoid (your defn).
Relevant:
http://en.wikipedia.org/wiki/Semiring http://en.wikipedia.org/wiki/Ring_(algebra) http://en.wikipedia.org/wiki/Ring_(algebra)#Notes_on_the_definition
/Joe
On Oct 7, 2009, at 5:41 PM, Daniel Fischer wrote:
Am Mittwoch 07 Oktober 2009 22:44:19 schrieb Joe Fredette:
A ring is an abelian group in addition, with the added operation (*) being distributive over addition, and 0 annihilating under multiplication. (*) is also associative. Rings don't necessarily need _multiplicative_ id, only _additive_ id. Sometimes Rings w/o ID is called a Rng (a bit of a pun).
/Joe
In my experience, the definition of a ring more commonly includes the multiplicative identity and abelian groups with an associative multiplication which distributes over addition are called semi-rings.
There is no universally employed definition (like for natural numbers, is 0 included or not; fields, is the commutativity of multiplication part of the definition or not; compactness, does it include Hausdorff or not; ...).