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Consider a noncommutative group ring over commuting polynomials

fricas
NPOLY:=MonoidRing(Polynomial(Fraction(Integer)),FreeGroup(Symbol))

\label{eq1}\hbox{\axiomType{MonoidRing}\ } (\hbox{\axiomType{Polynomial}\ } (\hbox{\axiomType{Fraction}\ } (\hbox{\axiomType{Integer}\ })) , \hbox{\axiomType{FreeGroup}\ } (\hbox{\axiomType{Symbol}\ }))(1)
Type: Type

Generators

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x:NPOLY:='x::FreeGroup Symbol

\label{eq2}x(2)
Type: MonoidRing?(Polynomial(Fraction(Integer)),FreeGroup?(Symbol))
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y:NPOLY:='y::FreeGroup Symbol

\label{eq3}y(3)
Type: MonoidRing?(Polynomial(Fraction(Integer)),FreeGroup?(Symbol))
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x':NPOLY:=('x::FreeGroup Symbol)^(-1)

\label{eq4}{x}^{- 1}(4)
Type: MonoidRing?(Polynomial(Fraction(Integer)),FreeGroup?(Symbol))
fricas
y':NPOLY:=('y::FreeGroup Symbol)^(-1)

\label{eq5}{y}^{- 1}(5)
Type: MonoidRing?(Polynomial(Fraction(Integer)),FreeGroup?(Symbol))
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a:POLY FRAC INT := 'a::Symbol

\label{eq6}a(6)
Type: Polynomial(Fraction(Integer))

Example

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p:=x*y*x+y

\label{eq7}y +{x \  y \  x}(7)
Type: MonoidRing?(Polynomial(Fraction(Integer)),FreeGroup?(Symbol))
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(p+3*a+1)*x'

\label{eq8}{{\left({3 \  a}+ 1 \right)}\ {{x}^{- 1}}}+{x \  y}+{y \ {{x}^{- 1}}}(8)
Type: MonoidRing?(Polynomial(Fraction(Integer)),FreeGroup?(Symbol))




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