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skewPolynomialRing(Ring,RingElement,List) -- Defines a skew polynomial ring via a scaling factor

Description

This method constructs a skew polynomial ring with coefficient ring R and generators elements of L. The relations all have the form a*b - f*b*a where a and b are in L. If R is a Bergman coefficient ring, an NCGroebnerBasis is computed for B.

i1 : R = QQ[q]/ideal{q^4+q^3+q^2+q+1}

o1 = R

o1 : QuotientRing
 
i2 : A = skewPolynomialRing(R,promote(2,R),{x,y,z,w})

o2 = A

o2 : FreeAlgebraQuotient
 
i3 : x*y == 2*y*x

o3 = true
 
i4 : B = skewPolynomialRing(R,q,{x,y,z,w})

o4 = B

o4 : FreeAlgebraQuotient
 
i5 : x*y == q*y*x

o5 = true
 
i6 : Bop = oppositeRing B
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o6 = Bop

o6 : FreeAlgebraQuotient
 
i7 : y*x == q*x*y

o7 = true
 
i8 : C = skewPolynomialRing(QQ,2_QQ, {x,y,z,w})

o8 = C

o8 : FreeAlgebraQuotient
 
i9 : x*y == 2*y*x

o9 = true
 
i10 : D = skewPolynomialRing(QQ,1_QQ, {x,y,z,w})

o10 = D

o10 : FreeAlgebraQuotient
 
i11 : isCommutative C

o11 = false
 
i12 : isCommutative D

o12 = false
 
i13 : Cop = oppositeRing C
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o13 = Cop

o13 : FreeAlgebraQuotient
 
i14 : y*x == 2*x*y

o14 = true

See also

Ways to use this method:

The source of this document is in AssociativeAlgebras/doc.m2:1008:0.