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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 noncommutative polynomial ring in n variables with coefficients in the commutative ring R is the free associative, unital R-algebra on n generators, which is noncommutative when n > 1.
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.
The twisted polynomial ring {} is defined as the set of polynomials in the variable and coefficients in . It is endowed with a ring structure with the usual addition, but with a non-commutative multiplication that can be summarized with the relation τ x = x p τ {\displaystyle \tau x=x^{p}\tau } for x ∈ k {\displaystyle x\in k} .
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 ...
Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
The ring of polynomials in finitely-many variables over the integers or a field is Noetherian. Rings that are not Noetherian tend to be (in some sense) very large. Here are some examples of non-Noetherian rings: The ring of polynomials in infinitely-many variables, X 1, X 2, X 3, etc.
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 ...
A matrix polynomial equation is an equality between two matrix polynomials, which holds for the specific matrices in question. A matrix polynomial identity is a matrix polynomial equation which holds for all matrices A in a specified matrix ring M n (R).