Commutative ring

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In ring theory, a branch of abstract algebra, 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, then a×b=b×a.

The study of commutative rings is called commutative algebra.

Contents

1 Examples
2 Constructing commutative rings
3 Properties
4 General discussion

Examples

The most important example is the ring of integers with the two operations of addition and multiplication. Ordinary multiplication of integers is commutative. This ring is usually denoted Z in the literature to signify the German word Zahlen (numbers).
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 class of fields is a subclass of the class of commutative rings.
A simple example of a non-commutative ring is the set of all 2-by-2 matrices whose entries are real numbers. For example, the matrix multiplication

:[\begin1 & 1\\0 & 1\\\end\cdot\begin1 & 1\\1 & 0\\\end=\begin2 & 1\\1 & 0\\\end]

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

:[\begin1 & 1\\1 & 0\\\end\cdot\begin1 & 1\\0 & 1\\\end=\begin1 & 2\\1 & 1\\\end.]

If n is a positive integer, then the set Z_n of integers modulo n forms a commutative ring with n elements (see modular arithmetic).

If R is a 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 RX₁,...,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.

If P is an ordinary prime number, the set of integers within the P-adic numbers forms a commutative ring.

Constructing commutative rings

Given a commutative ring, one can use it to construct new rings, as described below.

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 RX₁,...,X_n over a commutative ring R is a commutative ring, called the power series ring in n variables over R.
If S is a subset of a commutative ring R consisting of non-zero divisors and having the property that it is multiplicatively closed, i.e., that whenever t and u are in S then so is their product tu, then the set of all formal fractions (r,s) where r is any element of R and s is any element of S forms a new commutative ring, provided we define addition, subtraction, multiplication, and equality on this new set the same way we do for ordinary fractions. The new ring is denoted R_S and called the localization of R at S. The penultimate example above is the localization of the ring of integers at the multiplicatively closed subset of odd integers. The field of rationals is the localization of the commutative ring of integers at the multiplicative set of non-zero integers.
If I is an ideal in a commutative ring R, the powers of I form topological neighborhoods of 0 which allow R to be viewed as a topological ring. R can then be completed with respect to this topology. For example, if k is a field, kX, the formal power series ring in one variable over k, is the completion of k[X], the polynomial ring in one variable over k, under the topology generated by the powers of the ideal generated by X.

Properties

All subrings and quotient rings of commutative rings are also commutative.
If f : R → S is an injective ring homomorphism (that is, a monomorphism) between rings R and S, and if S is commutative, then R must also be commutative, since f(a·b) = f(a)·f(b) = f(b)·f(a) = f(b·a).
Similarly, if f : R → S is a ring homomorphism between rings R and S, and if R is commutative, the image f(R) of R is also commutative; in particular, if f is surjective (and therefore an epimorphism), S must be commutative.

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 than in the general case.

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. This subject is significantly more difficult when the commutative ring is not a field and is usually called homological algebra. The set of ideals within a commutative ring R can be exactly characterized as the set of R-modules which are submodules of R.

Some authors (such as I. N. Herstein) omit the requirement that a ring have a multiplicative identity. These authors call rings which do have multiplicative identities unital rings, unitary rings, or rings with identity. Authors such as Bourbaki, who do require rings to have an identity, call algebraic objects which meet all the requirements of a ring except the identity requirement pseudo-rings or rngs.

An element a of a commutative ring (with identity) is called a unit if it possesses a multiplicative inverse, i.e., if there exists another element b of the ring (with b not necessarily distinct from a) so that ab = ba = 1. Every nonzero element of a field is a unit. Every element of a commutative local ring not contained in the maximal ideal is a unit.

A non-zero element a of a commutative ring is said to be a zero divisor if there exists another non-zero element b of the ring (b not necessarily distinct from a) so that ab = 0. A commutative ring with identity which possesses no zero divisors is called an integral domain since it closely resembles the integers in some ways.

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