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

- Function: normalElements
- Usage:
    `normalElements(f,n)`
- Inputs:
  - f, a ring map,
  - n, an integer, a homogeneous degree in which to search for normal elements
- Outputs:
  - a matrix,

## Description

A normal element x in a non-commutative ring 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 a ring map 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})

    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 = oreExtension(B,sigma,a)
    Warning:  F4 Algorithm not available over current coefficient ring or inhomogeneous ideal.
    Converting to Naive algorithm.

    o3 = C

    o3 : FreeAlgebraQuotient

    i4 : sigmaC = map(C,C,{y,z,w,x,a})

    o4 = map (C, C, {y, z, w, x, a})

    o4 : RingMap C <-- C

    i5 : normalElements(sigmaC,1)

    o5 = | a |

                 1      1
    o5 : Matrix C  <-- C

    i6 : normalElements(sigmaC,2)

    o6 = 0

                  1
    o6 : Matrix QQ  <-- 0

    i7 : normalElements(sigmaC * sigmaC,2)

    o7 = | a2 |

                 1      1
    o7 : Matrix C  <-- C

    i8 : normalElements(sigmaC * sigmaC * sigmaC, 3)

    o8 = | a3 |

                 1      1
    o8 : Matrix C  <-- C

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/AssociativeAlgebras/doc.m2:280:0.