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

## Synopsis

• Usage:
isLeftRegular(x,n) or isRightRegular(x,n)
• Inputs:
• x, ,
• n, an integer, the degree in which regularity is checked.
• Outputs:

## 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}) Using numthreads = 0 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) Using numthreads = 0 o3 = true i4 : C = QQ<|x,y|> o4 = C o4 : FreeAlgebra i5 : D = C/ideal{x^2+x*y,y^2} Using numthreads = 0 o5 = D o5 : FreeAlgebraQuotient i6 : isLeftRegular(x,1) o6 = true i7 : isRightRegular(x,1) o7 = false