Difference between revisions of "Ring"
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* <math>(ab)c = a(bc)</math> (associativity of multiplication); | * <math>(ab)c = a(bc)</math> (associativity of multiplication); | ||
* For some <math>1\in R</math>, <math>1a=a1=a</math> (existance of multiplicative identity) | * For some <math>1\in R</math>, <math>1a=a1=a</math> (existance of multiplicative identity) | ||
− | * <math>a(b+c) | + | * <math>a(b+c)= ab+ac \\ (b+c)a = ba + ca = ab+ac </math> (double [[distributive property |distributivity]] of multiplication over addition). |
− | * <math>a(b-c) | + | * <math>a(b-c)= ab-ac \\ (b-c)a = ba - ca </math> (double [[distributive property |distributivity]] of multiplication over subtraction). |
Note especially that multiplicative inverses need not exist and that multiplication need not be commutative. | Note especially that multiplicative inverses need not exist and that multiplication need not be commutative. |
Latest revision as of 06:16, 8 April 2015
A ring is a structure of abstract algebra, similar to a group or a field. A ring is a set of elements closed under two operations, usually called multiplication and addition and denoted
and
, for which
is an abelian group;
is a monoid;
- Multiplication distributes doubly over addition.
In other words, the following properties hold for all in
:
(associativity of addition);
(commutativity of addition);
- For some
,
(existance of additive identity);
- There exists some
for which
(existance of additive inverses);
(associativity of multiplication);
- For some
,
(existance of multiplicative identity)
(double distributivity of multiplication over addition).
(double distributivity of multiplication over subtraction).
Note especially that multiplicative inverses need not exist and that multiplication need not be commutative.
The elements of under addition is called the additive group of
; it is sometimes denoted
. (However, this can sometimes lead to confusion when
is also an ordered set.) The set of invertible elements of
constitute a group under multiplication, denoted
. The elements of
under the multiplicative law
(i.e., the opposite multiplicative law) and the same additive law constitute the opposite ring of
, which can be denoted
.
Let be an element of
. Then the mapping
of
into
is an endomorphism of the abelian group
. Since group homomorphisms map identities to identities, it follows that
, for all
in
, and similarly,
.
Divisors
Let and
be elements of a ring
. If there exists an element
of
such that
, then
is said to be a right divisor of
, and
is said to be a left multiple of
. Left divisors and right multiples are defined similarly. When
is commutative, we say simply that
is a divisor of
, or
divides
, or
is a multiple of
.
Note that the relation " is a right divisor of
" is transitive, for if
and
, then
. Furthermore, every element of
is a right divisor of itself. Therefore
has the (sometimes trivial) structure of a partially ordered set.
Under these definitions, every element of is a left and right divisor of 0. However, by abuse of language, we usually only call an element
a left (or right) divisor of zero (or left, right zero divisors) if there is a non-zero element
for which
(or
). The left zero divisors are precisely those
elements of
for which left multiplication is not cancellable. For if
are distinct elements of
for which
, then
.
Examples of Rings
The sets of integers (), rational numbers (
), real numbers (
), and complex numbers (
) are all examples of commutative rings, as is the set of Gaussian integers (
). Note that of these, the integers and Gaussian integers do not have inverses; the rest do, and therefore also constitute examples of fields. All these rings are infinite, as well.
Among the finite commutative rings are sets of integers mod (
), for any integer
.
If is an abelian group, then the set of endomorphisms on
form a ring, under the rules
Let be a ring. The set of polynomials in
is also a ring.
Let be a field. The set of
matrices of
constitute a ring. In fact, they are the endomorphism ring of the additive group
.
If are rings, then Cartesian product
is a ring under coordinatewise multiplication and addition; this is called the direct product of these rings.
Let be the set of weak multiplicative functions mapping the positive integers into themselves. Then the elements of
form a pseudo-ring, with multiplication defined as Dirichlet convolution, i.e.,
for
However, there is no multiplicative identity, so this is not a proper ring.