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If R is a UFD, then so is R[X], the ring of polynomials with coefficients in R. Unless R is a field, R[X] is not a principal ideal domain. By induction, a polynomial ring in any number of variables over any UFD (and in particular over a field or over the integers) is a UFD.
This implies that, if R is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over R is a unique factorization domain. Another consequence is that factorization and greatest common divisor computation of polynomials with integers or rational coefficients may be ...
The definition of a polynomial ring can be generalised by relaxing the requirement that the algebraic structure R be a field or a ring to the requirement that R only be a semifield or rig; the resulting polynomial structure/extension R[X] is a polynomial rig.
The ring of Laurent polynomials over a field is Noetherian (but not Artinian). If R {\displaystyle R} is an integral domain , the units of the Laurent polynomial ring R [ X , X − 1 ] {\displaystyle R\left[X,X^{-1}\right]} have the form u X k {\displaystyle uX^{k}} , where u {\displaystyle u} is a unit of R {\displaystyle R} and k ...
Z[ω] (where ω is a primitive (non-real) cube root of unity), the ring of Eisenstein integers. Define f (a + bω) = a 2 − ab + b 2, the norm of the Eisenstein integer a + bω. K[X], the ring of polynomials over a field K. For each nonzero polynomial P, define f (P) to be the degree of P. [4] K[[X]], the ring of formal power series over the ...
If G is an abelian group, then the endomorphisms of G form a ring, the endomorphism ring End(G) of G. The operations in this ring are addition and composition of endomorphisms. More generally, if V is a left module over a ring R, then the set of all R-linear maps forms a ring, also called the endomorphism ring and denoted by End R (V).
More generally, the free algebra over any ring S may be used, and gives the concept of PI-algebra. If the degree of the polynomial P is defined in the usual way, the polynomial P is called monic if at least one of its terms of highest degree has coefficient equal to 1. Every commutative ring is a PI-ring, satisfying the polynomial identity XY ...
In mathematics, the ring of polynomial functions on a vector space V over a field k gives a coordinate-free analog of a polynomial ring. It is denoted by k[V]. If V is finite dimensional and is viewed as an algebraic variety, then k[V] is precisely the coordinate ring of V. The explicit definition of the ring can be given as follows.