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Encyclopedia > Commutative ring

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. In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have similar (but not identical) properties to those familiar from the integers. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if for all x and y in S. Otherwise, the operation * is noncommutative. ...


The study of commutative rings is called commutative algebra. In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ...


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.
  • 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

is not equal to the multiplication performed in the opposite order: In mathematics, an algebraic number relative to a field F is any element x of a given field K containing F such that x is a solution of a polynomial equation of the form anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the complex numbers are an extension of the real numbers by the inclusion of the imaginary unit i, satisfying . ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

  • If n is a positive integer, then the set Zn 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 R[[X1,...,Xn]] 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.

Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... The p-adic number systems were first described by Kurt Hensel in 1897. ...

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,...,Xn] 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,...,Xn]] 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 RS 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, k[[X]], 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.

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers like even number or multiple of 3. For instance, in rings one studies prime ideals instead of prime numbers, one defines coprime ideals as a generalization of... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ... In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. ... In mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as maps R × R → R, where R × R carries the product topology. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ...

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. In abstract algebra, a module is a generalization of a vector space. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...


Commutative rings are sometimes characterized by the elements they contain which have special properties. A multiplicative identity in a commutative ring is a special element (usually denoted 1) having the property that for every element a of the ring, 1×a = a. A commutative ring possessing such an element is said to be a ring with identity.


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 a×b = b×a = 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. In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of...


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 a×b = 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. In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ... In abstract algebra, an integral domain is a commutative ring with 0 ≠ 1 in which the product of any two non-zero elements is always non-zero. ...


  Results from FactBites:
 
Commutative ring - Wikipedia, the free encyclopedia (831 words)
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law.
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.
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.
Ring (mathematics) - Wikipedia, the free encyclopedia (1102 words)
Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings.
A ring (in the categorical sense) is commutative iff it is equal to its opposite ring.
The split-complex plane D is a ring useful in modern physics and is a subring of the tessarines.
  More results at FactBites »

 
 

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