oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring

Synopsis

Usage:

oreIdeal(A,sigma,delta,x) or oreIdeal(A,sigma,x)
Inputs:
- A, a ring,
- sigma, a ring map,
- delta, an instance of the type Derivation,
- x, a ring element, or a Symbol
Optional inputs:
- Degree => ..., default value 1
Outputs:
- an ideal,

Description

Given a ring A, an Ore extension of A by x is the quotient of the free extension A<x> by the relations x*a - sigma(a)*x-delta(a) where sigma is an automorphism of A and delta is a sigma-derivation. This method returns the defining ideal (in the appropriate tensor algebra) of an Ore extension of A by x. The current version assumes the sigma-derivation delta is 0, but this is to be fixed soon.

i1 : B = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w})
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o1 = B

o1 : FreeAlgebraQuotient

i2 : sigma = map(B,B,{y,z,w,x})

o2 = map (B, B, {y, z, w, x})

o2 : RingMap B <-- B

i3 : C = oreIdeal(B,sigma,a)

o3 = ideal (x*y + y*x, x*z + z*x, y*z + z*y, x*w + w*x, y*w + w*y, z*w + w*z,
     ------------------------------------------------------------------------
     - y*a + a*x, - z*a + a*y, - w*a + a*z, - x*a + a*w)

o3 : Ideal of QQ <|x, y, z, w, a|>

Ways to use oreIdeal :

oreIdeal(Ring,RingMap,Derivation,RingElement)
oreIdeal(Ring,RingMap,Derivation,Symbol)
oreIdeal(Ring,RingMap,RingElement)
oreIdeal(Ring,RingMap,Symbol)

For the programmer

The object oreIdeal is a method function with options.

oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring

Synopsis

Description

See also

Ways to use oreIdeal :

For the programmer