In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Mathematics, commutativity is the ability to change the order of something without changing the end result This means that if a and b are any elements of the ring, then ab = ba.

The study of commutative rings is called commutative algebra. Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings

Contents 1 Examples 2 Constructing commutative rings 3 Properties 4 General discussion 5 References

Examples

• The most important example is the ring of integers with the two operations of addition and multiplication. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French 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. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division
• 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. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally For example, the matrix multiplication

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

• 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). In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers
• If R is a given commutative ring, then the set of all polynomials in the variable X whose coefficients are in R forms a new commutative ring, denoted R[X]. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations
• Similarly, the set of formal power series R[[X₁,. 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 . . ,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 complete local ring. In Commutative algebra, the term completion refers to several related Functors on Topological rings and modules In Mathematics, more particularly in Abstract algebra, local rings are certain rings that are comparatively simple and serve to describe what is called
• 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 any prime number, the set of p-adic integers forms a commutative ring. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 In Mathematics, the p -adic number systems were first described by Kurt Hensel in 1897
• A set of matrices that can be diagonalized with the same similarity transformation forms a commutative ring. In Linear algebra, a Square matrix A is called diagonalizable if it is similar to a Diagonal matrix, i In Linear algebra, two n -by- n matrices A and B over the field K are called similar if there exists An example is the set of matrices of divided differences with respect to a fixed set of nodes. In Mathematics divided differences is a recursive division process

Constructing commutative rings

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

• Factor ring: 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. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the
• Localization: If S is a multiplicative subset of a commutative ring R then we can define the localization of R at S, or ring of fractions with denominators in S, usually denoted S^-1R. In Abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. In Abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. The penultimate example above is the localization of the ring of integers at the multiplicative 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.
• Completion: 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. In Commutative algebra, the term completion refers to several related Functors on Topological rings and modules In Mathematics, a topological ring is a ring R which is also a Topological space such that both the addition and the multiplication are This topology is called the I-adic topology. R can then be completed with respect to this topology. Formally, the I-adic completion is the inverse limit of the rings R/Iⁿ. In Mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects the precise For example, if k is a field, k[[X]], the formal power series ring in one variable over k, is the I-adic completion of k[X] where I is the principal ideal generated by X. 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 Analogously, the ring of p-adic integers is the I-adic completion of Z where I is the principal ideal generated by p.
• If R is a given commutative ring, the set of all polynomials R[X₁,. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations . . ,X_n] over R forms a new commutative ring, called the polynomial ring in n variables over R. In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables
• If Ris a given commutative ring, then the set of all formal power series R[[X₁,. 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 . . ,X_n]] over a commutative ring R is a commutative ring, called the power series ring in n variables over R.

Properties

• All subrings and quotient rings of commutative rings are also commutative. In Mathematics, a subring is a Subset of a ring, which contains the Multiplicative identity and is itself a ring under the same Binary operations In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring.
• 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). In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism.
• Similarly, if f : R → S is a ring homomorphism between rings R and S, and if R is commutative, the subring f(R) of S is also commutative; in particular, if f is surjective (and therefore an epimorphism), S must be commutative. In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which
• Every finite division ring is commutative (Wedderburn's theorem). In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible N. Jacobson^[1] has shown that the following condition is sufficient: if R is a ring such that for every element x of R there exists an integer n > 1 such that xⁿ = x, then R is commutative. Nathan Jacobson ( October 5, 1910 - December 5 1999) was an American Mathematician. Much more general conditions which guarantee commutativity of a ring were subsequently discovered by I. N. Herstein and others. Israel Nathan Herstein ( March 28, 1923, Lublin, Poland – February 9, 1988, Chicago, Illinois) was ^[2]

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 simplifies the situation considerably.

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. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars This subject is significantly more difficult when the commutative ring is not a field and is usually called homological algebra. Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting The set of ideals within a commutative ring R can be exactly characterized as the set of R-modules which are submodules of R.

An element a of a commutative ring (with identity) is called a unit if it possesses a multiplicative inverse, i. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i e. , if there exists another element b of the ring (with b not necessarily distinct from a) so that ab = 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 a non-zero element b of the ring such that ab = 0. In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such 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. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such

Some specific kinds of commutative rings are given with the following chain of inclusions:

commutative rings ⊃ integral domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields

Another possible chain (which is more geometric) is the following chain of inclusions:

Cohen-Macaulay rings ⊃ Gorenstein rings ⊃ Regular rings ⊃ Regular local rings

References

1. ^ Nathan Jacobson, Structure theory of algebraic algebras of bounded degree, Annals of Mathematics 46 (1945), no. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Mathematics, a unique factorization domain (UFD is roughly speaking a Commutative ring in which every element with special exceptions can be uniquely written In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i In Abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm applies In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Commutative algebra, a Gorenstein local ring is a Noetherian commutative Local ring R with finite Injective dimension, as an In Commutative algebra, a regular ring is a commutative Noetherian ring, such that the localization at every Maximal ideal is a Regular In Commutative algebra, a regular local ring is a Noetherian Local ring having the property that the minimal number of generators of its Maximal ideal 4, pp. 695–707. JSTOR
2. ^ James Pinter-Lucke, Commutativity conditions for rings: 1950–2005, Expositiones Mathematicae 25 (2007), no. 2, pp. 165–174. doi:10.1016/j.exmath.2006.07.001

• Atiyah M. F., Macdonald, I. G., Introduction to commutative algebra. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document. Sir Michael Francis Atiyah, OM, FRS, FRSE (b April 22, 1929) is a British Mathematician, and one of the Ian G Macdonald (born 1928 in London, England) is a British mathematician known for his contributions to Symmetric functions Special functions Addison-Wesley Publishing Co. , Reading, Mass. -London-Don Mills, Ont. 1969 ix+128 pp.
• Balcerzyk, Stanisław; Józefiak, Tadeusz, Commutative Noetherian and Krull rings. Translated from the Polish. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd. , Chichester; distributed by Prentice Hall, Inc. , Englewood Cliffs, NJ, 1989. 209 pp. ISBN 0-13-155615-0
• Balcerzyk, Stanisław; Józefiak, Tadeusz, Dimension, multiplicity and homological methods. Translated from the Polish. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd. , Chichester; distributed by Prentice Hall, Inc. , Englewood Cliffs, NJ, 1989. x+195 pp. ISBN 0-13-155623-1
• David Eisenbud, Commutative algebra. David Eisenbud (born 8 April, 1947) is an American Mathematician. With a view toward algebraic geometry. Graduate Texts in Mathematics, 150. Springer-Verlag, New York, 1995. xvi+785 pp. ISBN 0-387-94268-8; ISBN 0-387-94269-6 MR1322960
• Kaplansky, Irving, Commutative rings. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many Irving Kaplansky ( March 22, 1917 &ndash June 25, 2006) was a Canadian Mathematician. Revised edition. The University of Chicago Press, Chicago, Ill. -London, 1974. ix+182 pp.
• Nagata, Masayoshi, Local rings. Masayoshi Nagata ( Japanese: 永田 雅宜 Nagata Masayoshi; born 1927 in Aichi Japan (February 9 1927–August 27 2008 is a Japanese mathematician Interscience Tracts in Pure and Applied Mathematics, No. 13. Interscience Publishers a division of John Wiley and Sons, New York-London 1962 xiii+234 pp.
• Matsumura, Hideyuki, Commutative Ring Theory. Second edition. Translated from the Japanese. Cambridge Studies in Advanced Mathematics), Cambridge, UK : Cambridge University Press, 1989. ISBN 0-521-36764-6
• Zariski, Oscar; Samuel, Pierre, Commutative algebra. Oscar Zariski (born Oscher Zaritsky 24 April 1899 in Kobrin, Poland (today Belarus) died 4 July Pierre Samuel (born 12 September 1921 in Paris) is a French mathematician known for his work in Commutative algebra and its applications Vol. 1, 2. With the cooperation of I. S. Cohen. Corrected reprinting of the 1958, 1960 edition. Graduate Texts in Mathematics, No. 28, 29. Springer-Verlag, New York-Heidelberg-Berlin, 1975.

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