# normalElements(NCRingMap,ZZ) -- Finds elements normalized by a ring map

## Synopsis

• Function: normalElements
• Usage:
normalElements(f,n)
• Inputs:
• f, an instance of the type NCRingMap,
• n, an integer, a homogeneous degree in which to search for normal elements
• Outputs:

## Description

A normal element x in an NCRing R determines an automorphism f of R by a*x=x*f(a). Conversely, given a ring endomorphism, we may ask if any x satisfy the above equation for all a.

Given an NCRingMap f and a degree n, this method returns solutions to the equations a*x=x*f(a) for all generators a of R.

 i1 : B = skewPolynomialRing(QQ,(-1)_QQ,{x,y,z,w}) --Calling Bergman for NCGB calculation. Complete! o1 = B o1 : NCQuotientRing i2 : sigma = ncMap(B,B,{y,z,w,x}) o2 = NCRingMap B <--- B o2 : NCRingMap i3 : C = oreExtension(B,sigma,a) --Calling Bergman for NCGB calculation. Complete! o3 = C o3 : NCQuotientRing i4 : sigmaC = ncMap(C,C,{y,z,w,x,a}) o4 = NCRingMap C <--- C o4 : NCRingMap i5 : normalElements(sigmaC,1) o5 = | a | o5 : NCMatrix i6 : normalElements(sigmaC,2) o6 = 0 1 o6 : Matrix QQ <--- 0 i7 : normalElements(sigmaC @@ sigmaC,2) o7 = | a^2 | o7 : NCMatrix