Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms.[4] The proof makes use of the "1", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferable from the remaining rng assumptions only for elements that are products: ab + cd = cd + ab.)
Although most modern authors use the term "ring" as defined here, there are a few who use the term to refer to more general structures in which there is no requirement for multiplication to be associative.[5] For these authors, every algebra is a "ring".
Illustration
The integers, along with the two operations of addition and multiplication, form the prototypical example of a ring.
The most familiar example of a ring is the set of all integers
Z
,
{\displaystyle \mathbb {Z} ,} consisting of the numbers
…
,
−
5
,
−
4
,
−
3
,
−
2
,
−
1
,
0
,
1
,
2
,
3
,
4
,
5
,
…
{\displaystyle \dots ,-5,-4,-3,-2,-1,0,1,2,3,4,5,\dots }
The axioms of a ring were elaborated as a generalization of familiar properties of addition and multiplication of integers.
Some properties
Some basic properties of a ring follow immediately from the axioms:
The additive identity is unique.
The additive inverse of each element is unique.
The multiplicative identity is unique.
For any element x in a ring R, one has x0 = 0 = 0x (zero is an absorbing element with respect to multiplication) and (–1)x = –x.
If 0 = 1 in a ring R (or more generally, 0 is a unit element), then R has only one element, and is called the zero ring.
If a ring R contains the zero ring as a subring, then R itself is the zero ring.[6]
The binomial formula holds for any x and y satisfying xy = yx.