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 with coefficients in a ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of splitting fields, and to the understanding of a linear operator. Many important conjectures, such as Serre's conjecture, have influenced the study of other rings, and have influenced even the definition of other rings, such as group rings and rings of formal power series. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a polynomial is an expression that is constructed from one variable or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, Hilberts basis theorem, first proved by David Hilbert in 1888, states that, if k is a field, then every ideal in the ring of multivariate polynomials k[x1, x2, ..., xn] is finitely generated. ... In abstract algebra, the splitting field of a polynomial P(X) over a given field K is a field extension L of K, over which P factorizes into linear factors X − ai, and such that the ai generate L over K. It can be shown that such splitting fields exist... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described... 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...

Polynomials in one variable over a field

Polynomials

A polynomial in X with coefficients in a field K is an expression of the form 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. ...

where p₀, …, p_m are elements of K, the coefficients of p, and X, X ², … are formal symbols ("the powers of X"). Such expressions can be added and multiplied, and then brought into the same form using the ordinary rules for manipulating algebraic expressions, such as associativity, commutativity, distributivity, and collecting the similar terms. Any term p_kX ^k with zero coefficient, p_k = 0, may be omitted. The product of the powers of X is defined by the familiar formula In mathematics, associativity is a property that a binary operation can have. ... Example showing the commutativity of addition (3 + 2 = 2 + 3) For other uses, see Commute (disambiguation). ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ...

where k and l are any natural numbers. Two polynomials are considered to be equal if and only if the corresponding coefficients for each power of X are equal. By convention, X ¹ = X, X ⁰ = 1, and the sum defining the polynomial p may be viewed as the linear combination of the symbols X ^m, …, X ¹, X ⁰ with coefficients p_m, …, p₁, p₀. Using the summation symbol, the same polynomial is expressed more compactly as follows: In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... Sum redirects here. ...

The summation limits are frequently omitted, so that the same polynomial is written as

and it is understood that only finitely many terms are present, i.e. p_k is zero for all large enough values of k, in our case, for k > m. The degree of a polynomial is the largest k such that the coefficient of X ^k is not zero. In the special case of zero polynomial, all of whose coefficients are zero, the degree is undefined, or sometimes defined to be the symbol −∞ ("negative infinity").

The polynomial ring

The set of all polynomials with coefficients in K forms a commutative ring denoted K[X] and called the ring of polynomials over K. The symbol X is commonly called the "variable", and this ring is also called the ring of polynomials in one variable over K, to distiguish it from more general rings of polynomials in several variables. This terminology is suggested by the important cases of polynomials with real or complex coefficients, which may be alternatively viewed as real or complex polynomial functions. However, in general, X and its powers, X ^k, are treated as formal symbols, not as elements of the field K. One can think of the ring K[X] as arising from K by adding one new element X that is external to K and requiring that X commute with all elements of K. In order for K[X] to form a ring, all powers of X have to be included as well, and this leads to the definition of polynomials as linear combinations of the powers of X with coefficients in K. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ...

A ring has two binary operations, addition and multiplication. In the case of the polynomial ring K[X], these operations are explicitly given by the following formulas:

and

In the first formula, one of the polynomials may be extended by adding "dummy terms" with zero coefficients, so that the same set of powers formally appears in both summands. In the second formula, the inner summation in the right hand side is only extended over indices within bounds, 0 ≤ i ≤ m and 0 ≤ j ≤ n. Alternative forms of expressing addition and multiplication, without using explicit bounds in the sums, are as follows:

and

Since only finitely many coefficients a_i and b_j are non-zero, all sums in effect have only finitely many terms, and hence represent polynomials from K[X]. More generally, the field K can by replaced by any commutative ring R, giving rise to the polynomial ring over R , which is denoted R[X]. In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

Properties

The polynomial ring K[X] is remarkably similar to the ring Z of integers in many respects. This analogy and the arithmetic of the ring of polynomials were thoroughly investigated by Gauss and his theory served as a model for development of abstract algebra in the second half of the nineteenth century in the works of Kummer, Kronecker, and Dedekind. The integers are commonly denoted by the above symbol. ... Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (Gauß) (April 30, 1777 - February 23, 1855) was a legendary German mathematician, astronomer and physicist with a very wide range of contributions; he is considered to be one of the greatest mathematicians of all time. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the integers; all else is the work of man (Bell 1986, p. ... Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ...

K[X] is a domain

The first property of the polynomial ring is elementary and says that a product of two non-zero polynomials is also a non-zero polynomial. Indeed, the product of a polynomial p of degree m starting with p_mX ^m, p_m ≠ 0, and a polynomial q of degree n starting with q_nX ⁿ, q_n ≠ 0, is the polynomial pq starting with the term rX ^m+n, where the coefficient r = p_mq_n ≠ 0. Hence pq is a non-zero polynomial of degree m + n. Commutative rings in which the product of any two non-zero elements is non-zero are called integral domains, and thus the polynomial ring K[X] is an integral domain. In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ...

Factorization in K[X]

The next property of the polynomial ring is much deeper. Already Euclid noted that every positive integer can be uniquely factored into a product of primes — this statement is now called the fundamental theorem of arithmetic. The proof is based on Euclid's algorithm for finding the greatest common divisor of natural numbers. At each step of this algorithm, a pair (a, b), a > b, of natural numbers is replaced by a new pair (b, r), where r is the remainder from the division of a by b, and the new numbers are smaller. Gauss remarked that the procedure of division with the remainder can also be defined for polynomials: given two polynomials p and q, where q ≠ 0, one can write For other uses, see Euclid (disambiguation). ... In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself. ... In number theory, the fundamental theorem of arithmetic (or unique factorization theorem) states that every natural number either is itself a prime number, or can be written as a unique product of prime numbers. ... The Euclidean algorithm (also called Euclids algorithm) is an algorithm to determine the greatest common divisor (gcd) of two integers. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ...

p = uq + r,

where the quotient u and the remainder r are polynomials, the degree of r is less than the degree of q, and a decomposition with these properties is unique. The quotient and the remainer are found using the polynomial long division. The degree of the polynomial plays the role analogous to the size of an integer, and since it cannot decrease indefinitely, eventually, the division will be exact, and the last non-zero remainder is the greatest common divisor of the initial two polynomials. In this way, Gauss was able to simultaneously rigorously prove the fundamental theorem of arithmetic for integers and its generalization to polynomials. Commutative rings which admit an analogue of the Euclidean algorithm are called Euclidean rings, and for them, the unique factorization into prime factors holds, they are factorial rings, also called the unique factorization domains. Thus the polynomial ring K[X] is a factorial ring, which is moreover a Euclidean domain. In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of lower degree, a generalized version of the familiar arithmetic technique called long division. ... In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ... UFD redirects here, but this abbreviation can also mean USB flash drive, an electronic device. ... Wikipedia does not have an article with this exact name. ... In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ...

Another corollary of the polynomial division with the remainder is the fact that every non-zero, proper ideal I of K[X] is principal, i.e. I consists of the multiples of a single non-zero polynomial f which is the greatest common divisor for all polynomials in I. Thus the polynomial ring K[X] is a principal ideal domain. In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In Ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R. More specifically: a left principal ideal of R is a subset of R of the form Ra := {ra : r in R... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ...

Universal property of the polynomial ring

The ring K[X] of polynomials over K is obtained from K by adjoining one element, X. It turns out that any commutative ring L containing K and generated as a ring by a single element in addition to K can be described using K[X]. In particular, this applies to finite field extensions of K. In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. ...

Suppose that a commutative ring L contains K and there exists an element θ of L such that the ring L is generated by θ over K. Thus any element of L is a linear combination of powers of θ with coefficients in K. Then there is a unique ring homomorphism φ from K[X] into L which does not effect the elements of K itself (it is the identity map on K) and maps each power of X to the same power of θ. Its effect on the general polynomial amounts to "replacing X with θ": In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

By the assumption, any element of L appears as the right hand side of the last expression for suitable m and elements a₀, …, a_m of K. Therefore, φ is surjective and L is a homomorphic image of K[X]. More formally, let Ker φ be the kernel of φ. It is an ideal of K[X] and by the first isomorphism theorem for rings, L is isomorphic to the quotient of the polynomial ring K[X] by the ideal Ker φ. Since the polynomial ring is a principal ideal domain, this ideal is principal: there exists a polynomial p∈K[X] such that In mathematics, a surjective function (or onto function or surjection) is a function with the property that all possible output values of the function are generated when the input ranges over all the values in the domain. ... The word kernel has several meanings in mathematics, some related to each other and some not. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In Ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R. More specifically: a left principal ideal of R is a subset of R of the form Ra := {ra : r in R...

A particularly important application is to the case when the larger ring L is a field. Then the polynomial p must be irreducible. Conversely, the primitive element theorem states that any finite separable field extension L/K can be generated by a single element θ∈L and the preceding theory then gives a concrete description of the field L as the quotient of the polynomial ring K[X] by a principal ideal generated by an irreducible polynomial p. As an illustration, the field C of complex numbers is an extension of the field R of real numbers generated by a single element i such that i² + 1 = 0. Accordingly, the polynomial X² + 1 is irreducible over R and 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. ... In mathematics, the adjective irreducible means that an object cannot be expressed as a product of at least two non-trivial factors in a given ring. ... In mathematics, a primitive element for an extension of fields L/K is an element ζ of L such that L = K(ζ), or in other words such that L is generated by ζ over K. This means that every element of L can be written as a quotient of... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

More generally, given a (not necessarily commutative) ring A containing K and an element a of A that commutes with all elements of K, there is a unique ring homomorphism from the polynomial ring K[X] to A that maps X to a:

This homomorphism is given by the same formula as before, but it is not surjective in general. The existense and uniqueness of such a homomorphism φ expresses a certain universal property of the ring of polynomials in one variable and explains ubiquity of polynomial rings in various questions and constructions of ring theory and commutative algebra. In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... 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. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ...

The polynomial ring in several variables

Polynomials

A polynomial in n variables X₁,…, X_n with coefficients in a field K is defined analogously to a polynomial in one variable, but the notation is more cumbersome. For any multi-index α = (α₁,…, α_n), where each α_i is a non-negative integer, let

The product X^α is called the monomial of multidegree α. A polynomial is a finite linear combination of monomials with coefficients in K:

and only finitely many coefficients p_α are different from 0. The degree of a monomial X^α, frequently denoted |α|, is defined as

and the degree of a polynomial p is the largest degree of a monomial occuring with non-zero coefficient in the expansion of p.

The polynomial ring

Polynomials in n variables with coefficients in K form a commutative ring denoted K[X₁,…, X_n], or sometimes K[X], where X is a symbol representing the full set of variables, X = (X₁,…, X_n), and called the polynomial ring in n variables. This ring plays fundamental role in algebraic geometry. Many results in commutative and homological algebra originated in the study of its ideals and modules over this ring. Algebraic geometry is a branch of mathematics which, as the name suggests, combines techniques of abstract algebra, especially commutative algebra, with the language and the problematics of geometry. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ...

Hilbert's Nullstellensatz

Main article: Hilbert's Nullstellensatz

A group of fundamental results concerning the relation between ideals of the polynomial ring K[X₁,…, X_n] and algebraic subsets of Kⁿ originating with David Hilbert is known under the name Nullstellensatz (literally: "theorem on the set of zeros"). Hilberts Nullstellensatz (German: theorem of zeros) is a theorem in algebraic geometry that relates varieties and ideals in polynomial rings over algebraically closed fields. ... In mathematics, an algebraic set over a field K is the set of solutions in Kn (n-tuples of elements of K, of a set of simultaneous equations P1(X1, ...,Xn) = 0 P2(X1, ...,Xn) = 0 and so on up to Pm(X1, ...,Xn) = 0 for some integer m. ... | name = David Hilbert | image = Hilbert1912. ...

• (Weak form, algebraically closed field of coefficients). Let K be an algebraically closed field. Then every maximal ideal m of K[X₁,…, X_n] has the form

• (Strong form). Let k be a field with algebraic closure K, I be an ideal in the polynomial ring k[X₁,…, X_n], and V(I) be the algebraic subset of Kⁿ defined by I. Suppose that f is a polynomial which vanishes at all points of V(I). Then some power of f belongs to the ideal I:

Using the notion of the radical of an ideal, the conclusion says that f belongs to the radical of I. As a corollary of this form of Nullstellensatz, there is a bijective correspondence between the radical ideals of K[X₁,…, X_n] for an algebraically closed field K and the algebraic subsets of the n-dimensional affine space Kⁿ. It arises from the map

The prime ideals of the polynomial ring correspond to irreducible subvarieties of Kⁿ.

In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with coefficients in , has a zero (root) in . ... In mathematics, more specifically in ring theory a maximal ideal is an ideal which is maximal (with respect to set inclusion) amongst all proper ideals, i. ... In ring theory, a branch of mathematics, the radical of an ideal is a kind of completion of the ideal. ... In mathematics, a bijection, bijective function, or one-to-one correspondence is a function that is both injective (one-to-one) and surjective (onto), and therefore bijections are also called one_to_one and onto. ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ... In mathematics, the concept of irreducible component is used to make formal the idea that a set such as defined by the equation XY = 0 is the union of the two lines X = 0 and Y = 0. ...

Properties of the ring extension R ⊂ R[X]

One of the basic techniques in commutative algebra is to relate properties of a ring with properties of its subrings. The notation R ⊂ S indicates that a ring R is a subring of a ring S. In this case S is called an overring of R and one speaks of a ring extension. This works particularly well for polynomial rings and allows one to establish many important properties of the ring of polynomials in several variables over a field, K[X₁,…, X_n], by induction in n. In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ...

Summary of the results

In the following properties, R is a commutative ring and S = R[X₁,…, X_n] is the ring of polynomials in n variables over R. The ring extension R ⊂ S can be built from R in n steps, by successively adjoining X₁,…, X_n. Thus it to establish each of the properties below, it is sufficient to consider the case n = 1.

• If R is an integral domain then the same holds for S.

• If R is a unique factorization domain then the same holds for S. The proof is based on the Gauss lemma.

• Hilbert's basis theorem: If R is a Noetherian ring, then the same holds for S.

• Suppose that R is a Noetherian ring of finite global dimension. Then

An analogous result holds for Krull dimension.

In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... UFD redirects here, but this abbreviation can also mean USB flash drive, an electronic device. ... In the theory of polynomials, Gausss lemma, named after Carl Friedrich Gauss, refers to two related statements: The first result states that the product of two primitive polynomials is also primitive (a polynomial is primitive if the greatest common divisor of its coefficients is 1). ... In mathematics, Hilberts basis theorem, first proved by David Hilbert in 1888, states that, if k is a field, then every ideal in the ring of multivariate polynomials k[x1, x2, ..., xn] is finitely generated. ... In abstract algebra, a Noetherian ring is a ring that satisfies the ascending chain condition on ideals. ... The global dimension of a ring (denoted gl dim()) is the supremum of the set of projective dimensions of all -modules. ... In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull (1899 - 1971), is defined to be the number of strict inclusions in a maximal chain of prime ideals. ...

Generalizations

Polynomial rings have be generalized in a great many ways, including polynomial rings with generalized exponents, power series rings, non-commutative polynomial rings, and skew-polynomial rings.

Generalized exponents

Main article: Monoid ring

A simple generalization only changes the set from which the exponents on the variable are drawn. The formulas for addition and multiplication make sense as long as one can add exponents: Xⁱ·X^j = X^i+j. A set for which addition makes sense (is closed and associative) is called a monoid. The set of functions from a monoid N to a ring R which are nonzero at only finitely many places can be given the structure of a ring known as R[N], the monoid ring of N with coefficients in R. The addition is defined component-wise, so that if c = a+b, then c_n = a_n + b_n for every n in N. The multiplication is defined as the Cauchy product, so that if c = a·b, then for each n in N, c_n is the sum of all a_ib_j where i, j range over all pairs of elements of N which sum to n. In abstract algebra, a monoid ring is a procedure which constructs a new ring from a given ring and a monoid. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ...

When N is commutative, it is convenient to denote the function a in R[N] as the formal sum:

and then the formulas for addition and multiplication are the familiar:

and

where the latter sum is taken over all i, j in N that sum to n.

Some authors such as (Lang 2002, II,§3) go so far as to take this monoid definition as the starting point, and regular single variable polynomials are the special case where N is the monoid of non-negative integers. Polynomials in several variables simply take N to be the direct product of several copies of the monoid of non-negative integers.

Several interesting examples of rings and groups are formed by taking N to be the additive monoid of non-negative rational numbers, (Osbourne 2000, §4.4).

Power series

Main article: Formal power series

Power series generalize the choice of exponent in a different direction by allowing infinitely many nonzero terms. This requires various hypotheses on the monoid N used for the exponents, to ensure that the sums in the Cauchy product are finite sums. Alternatively, a topology can be placed on the ring, and then one restricts to convergent infinite sums. For the standard choice of N, the non-negative integers, there is no trouble, and the ring of formal power series is defined as the set of functions from N to a ring R with addition component-wise, and multiplication given by the Cauchy product. The ring of power series can be seen as the completion of the polynomial ring. 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...

Non-commutative polynomial rings

Main article: Free algebra

For polynomial rings of more than one variable, the products X·Y and Y·X are simply defined to be equal. A more general notion of polynomial ring is obtained when the distinction between these two formal products is maintained. Formally, the polynomial ring in n non-commuting variables with coefficients in the ring R is the monoid ring R[N], where the monoid N is the free monoid on n letters, also known as the set of all strings over an alphabet of n symbols, with multiplication given by concatenation. Neither the coefficients nor the variables need commute amongst themselves, but the coefficients and variables commute with each other. In abstract algebra, a free algebra is the noncommutative analogue of a polynomial ring. ... In abstract algebra, a monoid ring is a procedure which constructs a new ring from a given ring and a monoid. ... In abstract algebra, the free monoid on a set A is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from A, with the binary operation of concatenation. ...

Just as the polynomial ring in n variables with coefficients in the commutative ring R is the free commutative R-algebra of rank n, the non-commutative polynomial ring in n variables with coefficients in the commutative ring R is the free associative, unital R-algebra on n generators, which is non-commutative when n > 1.

Differential and skew-polynomial rings

Main article: Ore extension

Other generalizations of polynomials are differential and skew-polynomial rings. In mathematics, the Ore condition in ring theory is a condition introduced by Oystein Ore, in connection with the question of extending beyond commutative rings the construction of a field of fractions, or more general localization of a ring. ...

A differential polynomial ring is formed from a ring R and a derivation δ of R into R. Then the multiplication is extended from the relation X·a = a·X + δ(a). The standard example, called a Weyl algebra, takes R to be a polynomial ring k[t], and X to be the standard polynomial derivative . One views the elements of R[X] as differential operators on the polynomial ring k[t], with elements f(t) of R=k[t] acting as multiplication, and X acting as the derivative in t. Labelling t = Y, one gets the canonical commutation relation, X·Y − Y·X = 1, making the ring explicitly a Weyl algebra. This is a fundamentally important ring, (Lam 2001, §1,ex1.9). In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), More precisely, let F be a field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F[X]. ∂X... In physics, the canonical commutation relation is the relation among the position and momentum of a point particle in one dimension, where is the so-called commutator of and , is the imaginary unit and is the reduced Plancks constant . ... In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), More precisely, let F be a field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F[X]. ∂X...

The skew-polynomial ring is defined for a ring R and a ring endomorphism f of R, multiplication is extended from the relation X·r = f(r)·X to give an associative multiplication that distributes over the standard addition. More generally, one has a homomorphism F from the monoid N into the endomorphism ring of R, and Xⁿ·r = F(n)(r)·Xⁿ, as in (Lam 2001, §1,ex 1.11). Skew polynomial rings are closely related to crossed product algebras. In mathematics, and more specifically in the theory of von Neumann algebras, a crossed product is a basic method of constructing a new von Neumann algebra from a von Neumann algebra acted on by a group. ...

See also

• Additive polynomial
• Laurent polynomial

In mathematics, the additive polynomials are an important topic in classical algebraic number theory. ... In mathematics, a Laurent series is an infinite series. ...

References

• Lam, Tsit-Yuen (2001), A First Course in Noncommutative Rings, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95325-0
• Lang, Serge (2002), Algebra, vol. 211, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR1878556, ISBN 978-0-387-95385-4
• Osborne, M. Scott (2000), Basic homological algebra, vol. 196, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, MR1757274, ISBN 978-0-387-98934-1