Ring (mathematics)

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 and the ring of matrices.

Formal definition

A ring is a set R equipped with two binary operations, which are generally denoted + and · and called addition and multiplication, respectively, such that:

• (R, +) is an abelian group
• Multiplication is associative
• The left and right distributive laws hold:
  • a·(b + c) = (a·b) + (a·c)
  • (a + b)·c = (a·c) + (b·c)

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).

Types of Rings

• 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.

• A ring in which the multiplication operation is commutative is called a commutative ring: for example, the integers. Such commutative rings are the basic object of study in commutative algebra, in which rings are generally also assumed to have a unit. An example of a ring in which multiplication is not commutative is the set of all 2-by-2 matrices with integer coefficients. By the definition of rings, addition is necessarily commutative.

• A unital ring in which every 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

A ring homomorphism is a mapping π from a ring A to a ring B respecting the ring operations. That is,

• π(ab) = π(a)π(b)
• π(a+b) = π(a) + π(b)

If the rings are unital, it is often assumed that π maps the identity element of A to the identity element of B.

Subrings

If A is a ring, a subset B of A is called a subring if B is a ring under the ring operations inherited from A. It can be seen that this is equivalent to requiring that B is closed under multiplication and subtraction.

If A is unital, some authors demand that a subring of A should contain the unit of A.

Ideals

A two-sided ideal of a ring A is a subring I such that for any element a in A and any element b in I we have that ab and ba are elements of I. 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 A by declaring that two elements of A are equivalent if their difference is an element of I. The set of equivalence classes is then denoted by A/I and is a ring with the induced operations.

If h:A→B is a ring homomorphism, then the inverse image of 0, { x∈A: h(x)=0 }, is an ideal of A. Conversely, if I is an ideal of A, then there is a natural ring homomorphism from A to A/I such that I is the set of all elements mapped to 0 in A/I.

Examples

• The trivial ring {0} consists of only one element, which serves ar both additive and multiplicative identity.
• The integers forms a ring with addition and multiplication defined as usual. This is a commutative ring.
  • The rational, real and complex numbers all form commutative rings.
• The set of polynomials forms a commutative ring.
• The set of square $n\times n$ 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

• For every ring R we can define the opposite ring R^op by reversing the multiplication in R. Given the multiplication · in R the multiplication ∗ in R^op is defined as b∗a := a·b. The "identity map" from R to R^op is an isomorphism if and only if R is commutative. However, even if R is not commutative, it is still possible for R and R^op to be isomorphic. For example, if R is the ring of n×n matrices of real numbers, then the transposition map from R to R^op is an isomorphism.
• The center of a ring R is the set of elements of R that commute with every element of R; that is, c lies in the center if cr=rc for every r in R. The center is a subring of R. We say that a subring S of R is central if it is a subring of the center of R.
• 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 $\{R_{j}\}_{j\in J}$, the direct product and direct sum exist. The direct product is the collection of "infinite-tuples" $\{r_{j}\}_{j\in J}$ with component-wise addition and multiplication as operations.
• The direct sum of a collection of rings $\{R_{j}\}_{j\in J}$ is the subring of the direct product consisting of all infinite-tuples $\{r_{j}\}_{j\in J}$ with the property that r_j=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

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

• Glossary of ring theory
• Algebra over a commutative ring
• Nonassociative ring

• Special types of rings:
  • Commutative ring
  • Differential ring
  • Division ring
  • Field
  • Integral domain (ID)
  • Principal ideal domain (PID)
  • Unique factorization domain (UFD)

References

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