Commutative ring

In mathematics, a commutative ring is a ring in which the multiplication operation obeys the commutative law. This means that if a and b are any elements of the ring, and if the multiplication operation is written as *, then a*b=b*a.

The study of commutative rings is called commutative algebra.

History

See Ring theory

Examples

• The most important example is the ring of integers with the two operations of addition and multiplication. Ordinary multiplication of integers is commutative.
• The rational, real and complex numbers form commutative rings, in fact they are even fields.
• More generally, every field is a commutative ring, so the set of fields is a subset of the set of commutative rings.
• The easiest example of a non-commutative ring is the set of all square 2-by-2 matrices whose entries are real numbers. For example, the matrix multiplication

${\begin{bmatrix}1&1\\0&1\\\end{bmatrix}}\cdot {\begin{bmatrix}1&1\\1&0\\\end{bmatrix}}={\begin{bmatrix}2&1\\1&0\\\end{bmatrix}}$

is not equal to the multiplication performed in the opposite order:

${\begin{bmatrix}1&1\\1&0\\\end{bmatrix}}\cdot {\begin{bmatrix}1&1\\0&1\\\end{bmatrix}}={\begin{bmatrix}1&2\\1&1\\\end{bmatrix}}$

• If n is a positive integer, then the set Z_n of integers modulo n forms a ring with n elements (see modular arithmetic).
• If R is given commutative ring, then the set of all polynomials in the variable X whose coefficient are from R forms a new commutative ring, denoted R[X].
• Similarly, the set of formal power series R[[X₁,...,X_n]] over a commutative ring R is a commutative ring. If R is a field the formal power series ring is a special kind of commutative ring, called a local ring.
• The set of all ordinary rational numbers whose denominator is odd forms a commutative ring, in fact a local ring. This ring contains the ring of integers properly, and is itself a proper subset of the rational field.

Constructing new commutative rings from given ones

• Given a commutative ring R and an ideal I of R, the 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.
• If R is a given commutative ring, the set of all polynomials R[X₁,...,X_n] over R forms a new commutative ring, called the polynomial ring in n variables over R.
• If R is given commutative ring, then the set of all formal power series R[[X₁,...,X_n]] over a commutative ring R is a commutative ring, called the power series ring in n variables over R.

General Discussion

The inner structure of a commutative ring is determined by considering its ideals. All ideals in a commutative ring are two-sided, which makes considerations considerably easier.

The outer structure of a commutative ring is determined by considering linear algebra over that ring, i.e., by investigating the theory of its modules.