Joe Fredette wrote:
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.
Yes, this is how I learned it in my Algebra course(*). Though I can imagine that this is not universally agreed upon; indeed most of the more interesting results need a multiplicative unit, IIRC, so there's a justification for authors to include it in the basic definition (so they don't have to say let-R-be-a-ring-with-multiplicative-unit all the time ;-) Cheers Ben (*) As an aside, this was given by one Gernot Stroth, back then still at the FU Berlin, of whom I later learned that he took part in the grand effort to classify the simple finite groups. The course was extremely tight but it was fun and I never again learned so much in one semester.