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isLeftRegular -- Determines if a given (homogeneous) element is regular in a given degree

Synopsis

Description

Given an element x in an noncommutative ring, isLeftRegular returns true if a*x=0 implies a=0 for all a in the specified homogeneous degree n. Likewise isRightRegular returns true if x*a=0 implies a=0 for all elements a of degree n. The method calls leftMultiplicationMap or rightMultiplicationMap as appropriate and checks the kernel in the specified degree.

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

o1 = B

o1 : FreeAlgebraQuotient
 
i2 : g = z^3 + y*z*x - z*y*x - y^3

        3                    3
o2 = - y  + y*z*x - z*y*x + z

o2 : B
 
i3 : isLeftRegular(g,6)
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o3 = true
 
i4 : C = QQ<|x,y|>

o4 = C

o4 : FreeAlgebra
 
i5 : D = C/ideal{x^2+x*y,y^2}
Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
Converting to Naive algorithm.

o5 = D

o5 : FreeAlgebraQuotient
 
i6 : isLeftRegular(x,1)

o6 = true
 
i7 : isRightRegular(x,1)

o7 = false

See also

Ways to use isLeftRegular :

For the programmer

The object isLeftRegular is a method function.