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; ...).