In mathematics, a ring is an algebraic structure with two binary operations, commonly called addition and multiplication. These operations are defined so as to emulate and generalize the integers. Other common examples of rings include the ring of polynomials of one variable with real coefficients, or a ring of square matrices of a given dimension.
To qualify as a ring, addition must be commutative and each element must have an inverse under addition: for example, the additive inverse of 3 is -3. However, multiplication in general does not satisfy these properties. A ring in which multiplication is commutative and every element except the additive identity element (0) has a multiplicative inverse (reciprocal) is called a field: for example, the set of rational numbers. (The only ring in which 0 has an inverse is the trivial ring of only one element.)
A ring can have a finite or infinite number of elements. An example of a ring with a finite number of elements is
, the set of remainders when an integer is divided by 5, i.e. the set {0,1,2,3,4} with operations such as 4 + 4 = 3 because 8 has remainder 3 when divided by 5. A similar ring
can be formed for other positive values of
.
Formal definition[edit]
A ring is a set R equipped with two binary operations, which are generally denoted + and · and called addition and multiplication, respectively, such that:
In practice, the symbol · is usually omitted, and multiplication is just denoted by juxtaposition. The usual order of operations is also assumed, so that a + bc is an abbreviation for a + (b·c). The distributive property is specified separately for left and right multiplication to cover cases where multiplication is not commutative, such as a ring of matrices.
Types of rings[edit]
Unital ring[edit]
A ring in which there is an identity element for multiplication is called a unital ring, unitary ring, or simply ring with identity. The identity element is generally denoted 1. Some authors, notably Bourbaki, demand that their rings should have an identity element, and call rings without an identity pseudorings.
Commutative ring[edit]
A ring in which the multiplication operation is commutative is called a commutative ring. Such commutative rings are the basic object of study in commutative algebra, in which rings are generally also assumed to have a unit.
Division ring[edit]
- For more information, see: Division ring.
A unital ring in which every non-zero element a has an inverse, that is, an element a−1 such that a−1a = aa−1 = 1, is called a division ring or skew field.
Homomorphisms of rings[edit]
A ring homomorphism is a mapping
from a ring
to a ring
respecting the ring operations. That is,
![{\displaystyle \pi (ab)=\pi (a)\pi (b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91de980593be752d23eefdbf03df1fd145922d31)
![{\displaystyle \pi (a+b)=\pi (a)+\pi (b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35168edd1df880947d88201d4ce61f7813a5f6b6)
If the rings are unital, it is often assumed that
maps the identity element of
to the identity element of
.
A homomorphism can map a larger set onto a smaller set; for example, the ring
could be the integers
and could be mapped onto the trivial ring which contains only the single element
.
Subrings[edit]
If
is a ring, a subset
of
is called a subring if
is a ring under the ring operations inherited from
. It can be seen that this is equivalent to requiring that
be closed under multiplication and subtraction.
If
is unital, some authors demand that a subring of
should contain the unit of
.
A two-sided ideal of a ring
is a subring
such that for any element
in
and any element
in
we have that
and
are elements of
. The concept of ideal of a ring corresponds to the concept of normal subgroups of a group. Thus, we may introduce an equivalence relation on
by declaring that two elements of
are equivalent if their difference is an element of
. The set of equivalence classes is then denoted by
and is a ring with the induced operations.
If
is a ring homomorphism, then the kernel of h, defined as the inverse image of 0,
, is an ideal of
. Conversely, if
is an ideal of
, then there is a natural ring homomorphism, the quotient homomorphism, from
to
such that
is the set of all elements mapped to 0 in
.
Examples[edit]
- The trivial ring {0} consists of only one element, which serves as both additive and multiplicative identity.
- The integers form a ring with addition and multiplication defined as usual. This is a commutative ring.
- The set of polynomials forms a commutative ring.
- The set of square
matrices forms a ring under componentwise addition and matrix multiplication. This ring is not commutative if n>1.
- The set of all continuous real-valued functions defined on the interval [a,b] forms a ring under pointwise addition and multiplication.
Constructing new rings from given ones[edit]
- For every ring
we can define the opposite ring
by reversing the multiplication in
. Given the multiplication
in
, the multiplication
in
is defined as
. The "identity map" from
to
, mapping each element to itself, is an isomorphism if and only if
is commutative. However, even if
is not commutative, it is still possible for
and
to be isomorphic using a different map. For example, if
is the ring of
matrices of real numbers, then the transposition map from
to
, mapping each matrix to its transpose, is an isomorphism.
- The center of a ring
is the set of elements of
that commute with every element of
; that is,
is an element of the center if
for every
. The center is a subring of
. We say that a subring
of
is central if it is a subring of the center of
.
- The direct product of two rings R and S is the cartesian product R×S together with the operations
- (r1, s1) + (r2, s2) = (r1+r2, s1+s2) and
- (r1, s1)(r2, s2) = (r1r2, s1s2).
- With these operations R×S is a ring.
- More generally, for any index set J and collection of rings
, the direct product and direct sum exist.
- The direct product is the collection of "infinite-tuples"
with component-wise addition and multiplication as operations.
- The direct sum of a collection of rings
is the subring of the direct product consisting of all infinite-tuples
with the property that rj=0 for all but finitely many j. In particular, if J is finite, then the direct sum and the direct product are isomorphic, but in general they have quite different properties.
- Since any ring is both a left and right module over itself, it is possible to construct the tensor product of R over a ring S with another ring T to get another ring, provided S is a central subring of R and T.
History[edit]
The study of rings originated from the study of polynomial rings and algebraic number fields in the second half of the nineteenth century, amongst other by Richard Dedekind. The term ring itself, however, was coined by David Hilbert in 1897.
See also[edit]
- Rings with added structure
References[edit]
Fraleigh, John B. 2003. A First Course in Abstract Algebra. 7th ed. Boston: Addison-Wesley
Hilbert, David. 1897. Die Theorie der algebraische Zahlkoerper, ahresbericht der Deutschen Mathematiker Vereiningung vol. 4.
Lang, Serge. 2002. Algebra. 3rd ed. New York: Springer