Definition
In mathematics a ring is a commutative group (R, +), together with a second binary operation * such that for all a, b and c in R,
- a * (b * c) = (a * b) * c
- a * (b + c) = (a * b) + (a * c)
- (a + b) * c = (a * c) + (b * c)
and such that there exists a multiplicative identity, or unity, that is, an element 1 such that for all a in R,
- a * 1 = 1 * a = a.
Some authors omit the requirement for a multiplicative identity, and call those rings which do have multiplicative identities unitary rings. In this encyclopedia, the existence of a multiplicative identity is taken to be part of the definition.
Examples
- The motivating example is the ring of integers with the two operations of addition and multiplication.
- The rational, real and complex numbers form rings, in fact they are even fields.
- If n is a positive integer, then the set Zn of integers modulo n forms a ring with n elements (see modular arithmetic).
- The set of all continuous real-valued functions defined on the interval [a, b] forms a ring (even an associative algebra). The operations are addition and multiplication of functions.
- The set of all square n-by-n matrices, where n is fixed, forms a ring with matrix addition and matrix multiplication as operations.
- The set of all polynomials over some common coefficient ring forms a ring.
- If G is an abelian group, then the endomorphisms form a ring, the endomorphism ring End(G) of G. The operations in this ring are addition and composition of endomorphisms.
First consequences
From the axioms, one can immediately deduce that
- 0 * a = a * 0 = 0
- (-1) * a = -a
- (-a) * b = a * (-b) = -(a * b)
for all elements
a and
b in
R. Here, 0 is the neutral element with respect to addition +, and -
x stands for the additive inverse of the element
x in
R.
An element a in a ring is called invertible if there is an element b such that
- a * b = b * a = 1.
If that is the case,
b is uniquely determined by
a and we write
a-1 =
b.
The set of all invertible elements in a ring is closed under multiplication * and therefore forms a
group, the
group of units of the ring. If both
a and
b are invertible, then we have
- (a * b)-1 = b-1 * a-1
Special types of rings
A ring is called commutative if its multiplication is commutative.
A commutative ring with more than one element where no two non-zero elements multiply to give zero is called an [integral domain]?. In such rings, multiplicative cancellation is possible.
Of particular interest are fields, integral domains where every non-zero element is invertible.
Some important concepts: subring and ring ideal, module, field, semigroup ring
/Talk
?
![[Home]](wikipedia.png)