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. 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).
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; ...).
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