Ring (mathematics) - Definition and Illustration

Definition and Illustration

The most familiar example of a ring is the set of all integers, Z, consisting of the numbers

. . ., −4, −3, −2, −1, 0, 1, 2, 3, 4, . . .

It serves as a prototype for the axioms for rings. A ring is a set R equipped with two binary operations + : R × R → R and · : R × R → R (where × denotes the Cartesian product), called addition and multiplication, such that:

• (R, +) is an abelian group with identity element 0, meaning that for all a, b, c in R, the following axioms hold:
  • (a + b) + c = a + (b + c) (+ is associative)
  • 0 + a = a (0 is the identity)
  • a + b = b + a (+ is commutative)
  • for each a in R there exists −a in R such that a + (−a) = (−a) + a = 0 (−a is the inverse element of a)
• (R, ·) satisfies
  • (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) (⋅ is associative)
• Multiplication distributes over addition:
  • a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c)
  • (a + b) ⋅ c = (a ⋅ c) + (b ⋅ c).

As with groups the symbol ⋅ is usually omitted and multiplication is just denoted by juxtaposition.

Although ring addition is commutative, so that a + b = b + a, ring multiplication is not required to be commutative; a ⋅ b need not equal b ⋅ a. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.

Some basic properties of a ring follow immediately from the axioms.

• The additive identity and the the additive inverse are unique.
• The binomial formula holds for any commuting elements (i.e., ).

Authors do not agree on whether a ring has the multiplicative identity 1 or not; i.e., an element such that . For example, the set of even integers satisfies above axioms and thus constitutes a ring but does not have 1. Rings which do have multiplicative identities are sometimes for emphasis referred to as unital rings, unitary rings, rings with unity, rings with identity or rings with 1. The present article does not go in depth on this issue. The interested readers are referred to the article pseudo-rings.

Following the common practices in Wikipedia, this article adopts the following convention: a commutative ring is assumed to have the identity, while a ring in general is not assumed so. To discuss a commutative ring that is not necessarily unital, we will use the term commutative Z-algebra.