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 polynomials and 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 and identity element, and call rings without an identity pseudorings.

• 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.

• 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.

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.

Basic theorems

From the axioms, one can immediately deduce that, for all elements a and b of a ring, we have

• 0a = a0 = 0.
• (−1)a = −a.
• (−a)b = a(−b) = −(ab).
• (ab)⁻¹ = b⁻¹ a⁻¹ if both a and b are invertible.

Other basic theorems

• The identity element 1 is unique.
• If the ring has at least two elements then 0 ≠ 1
• The binomial theorem

$(x+y)^{n}=\sum _{k=0}^{n}{n \choose k}x^{k}y^{n-k},$

holds in any commutative ring.

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.
• If a subset S of a ring R is closed under multiplication and subtraction and contains the multiplicative identity element, then S is called a subring of R.
• 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).

• More generally, for any index set J and collection of rings (R_j)_jεJ, the direct product and direct sum exist. The direct product is the collection of "infinite-tuples" (r_j)_jεJ with component-wise addition and multiplication. More formally, let U be the union of all of the rings R_j. Then the direct product of the R_j over all jεJ is the set of all maps r:J→U with the property that r_jεR_j. Addition and multiplication of these functions is via the addition and multiplication in each individual R_j. Thus

(r+s)_j=r_j+s_j and (rs)_j=r_js_j.

• The direct sum of a collection of rings (R_j)_jεJ is the subring of the direct product consisting of all infinite-tuples (r_j)_jεJ 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.
• Given a ring R and an ideal I of R, the quotient ring (or factor ring) R/I is the set of cosets of I together with the operations

(a+I) + (b+I) = (a+b) + I and

(a+I)(b+I) = (ab) + I.

• 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.

See also

• Ring theory
• 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)