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In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials k[x 11,...,x dn] generated by the d-by-d minors of a generic d-by-n matrix (x ij). The bracket ring may be regarded as the ring of polynomials on the image of a Grassmannian under the Plücker embedding. [1]
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 ...
In ring theory, the commutator [a,b] is defined as ab − ba. Furthermore, braces may be used to denote the anticommutator: {a,b} is defined as ab + ba. The Lie bracket of a Lie algebra is a binary operation denoted by [,]:. By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra.
In the mathematical field of knot theory, the bracket polynomial (also known as the Kauffman bracket) is a polynomial invariant of framed links.Although it is not an invariant of knots or links (as it is not invariant under type I Reidemeister moves), a suitably "normalized" version yields the famous knot invariant called the Jones polynomial.
The bracket polynomial is known to change by a factor of ... is the ring of Laurent polynomials with integer coefficients in the variable / . Definition by braid ...
The Jones polynomial is a special case of the Kauffman polynomial, as the L polynomial specializes to the bracket polynomial. The Kauffman polynomial is related to Chern–Simons gauge theories for SO(N) in the same way that the HOMFLY polynomial is related to Chern–Simons gauge theories for SU(N). [2]
In mathematics Hilbert's basis theorem asserts that every ideal of a polynomial ring over a field has a finite generating set (a finite basis in Hilbert's terminology). In modern algebra, rings whose ideals have this property are called Noetherian rings. Every field, and the ring of integers are Noetherian rings.
Sometimes the term noncommutative ring is used instead of ring to refer to an unspecified ring which is not necessarily commutative, and hence may be commutative. Generally, this is for emphasizing that the studied properties are not restricted to commutative rings, as, in many contexts, ring is used as a shorthand for commutative ring.