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; ...). _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe