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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.
Strictly the X i here are "non-commuting indeterminates", and so "polynomial identity" is a slight abuse of language, since "polynomial" here stands for what is usually called a "non-commutative polynomial". The abbreviation PI-ring is common. More generally, the free algebra over any ring S may be used, and gives the concept of PI-algebra. If ...
The Weyl algebras are Ore extensions, with R any commutative polynomial ring, σ the identity ring endomorphism, and δ the polynomial derivative. Ore algebras are a class of iterated Ore extensions under suitable constraints that permit to develop a noncommutative extension of the theory of Gröbner bases.
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} .
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.
In algebra, a λ-ring or lambda ring is a commutative ring together with some operations λ n on it that behave like the exterior powers of vector spaces.Many rings considered in K-theory carry a natural λ-ring structure. λ-rings also provide a powerful formalism for studying an action of the symmetric functions on the ring of polynomials, recovering and extending many classical results ...
The ring of symmetric functions is a convenient tool for writing identities between symmetric polynomials that are independent of the number of indeterminates: in Λ R there is no such number, yet by the above principle any identity in Λ R automatically gives identities the rings of symmetric polynomials over R in any number of indeterminates ...