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The skew-polynomial ring is defined similarly for a ring R and a ring endomorphism f of R, by extending the multiplication from the relation X⋅r = f(r)⋅X to produce an associative multiplication that distributes over the standard addition.
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.
A formal power series ring does not have the universal property of a polynomial ring; a series may not converge after a substitution. The important advantage of a formal power series ring over a polynomial ring is that it is local (in fact, complete).
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 ...
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 dimension of the graded ring = / + (equivalently, 1 plus the degree of its Hilbert polynomial). A commutative ring R is said to be catenary if for every pair of prime ideals ′, there exists a finite chain of prime ideals = = ′ that is maximal in the sense that it is impossible to insert an additional prime ideal between two ideals ...
As the latter formulation makes clear, a polynomial ring is the group algebra over a vector space, and writing in terms of corresponds to choosing a basis for the vector space. Then an ideal I, or equivalently a module R / I , {\displaystyle R/I,} is a cyclic representation of R (cyclic meaning generated by 1 element as an R -module; this ...
Every field, and the ring of integers are Noetherian rings. So, the theorem can be generalized and restated as: every polynomial ring over a Noetherian ring is also Noetherian. The theorem was stated and proved by David Hilbert in 1890 in his seminal article on invariant theory [1], where he solved several problems on invariants.