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The ring of formal power series over the complex numbers is a UFD, but the subring of those that converge everywhere, in other words the ring of entire functions in a single complex variable, is not a UFD, since there exist entire functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must 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.
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).
The formal power series over a ring form a ring, commonly denoted by [[]]. (It can be seen as the (x)-adic completion of the polynomial ring [], in the same way as the p-adic integers are the p-adic completion of the ring of the integers.)
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 ...
Any nonzero polynomial may be decomposed, in a unique way, as a sum of homogeneous polynomials of different degrees, which are called the homogeneous components of the polynomial. Given a polynomial ring = [, …,] over a field (or, more generally, a ring) K, the homogeneous polynomials of degree d form a vector space (or a module), commonly ...
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 ...
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 ...