Under the hood these are the sort of calculations that my code is doing:
{{{
sage: Rq.<q> = LaurentPolynomialRing(ZZ,'q')
sage: Ruq.<u> = PolynomialRing(Rq,'u')
sage: mat = matrix([[q-q,u-q],[1,1]])
Under the hood these are the sort of calculations that my code is doing:

sage: Rq.<q> = LaurentPolynomialRing(ZZ,'q')
sage: Ruq.<u> = PolynomialRing(Rq,'u')
sage: mat = matrix([[q-q,u-q],[1,1]])
sage: mat.rref()
[              1               0]
[              0 (u - q)/(u - q)]
sage: c = _[1][1]; c; c.parent()
(u - q)/(u - q)
Fraction Field of Univariate Polynomial Ring in u over Univariate Laurent 
Polynomial Ring in q over Integer Ring
sage: c==1
True
sage: c.reduce()
-------------------------------------  ------------------------------
ArithmeticError                           Traceback
...
ArithmeticError: unable to reduce because gcd algorithm not implemented on 
input

It may be an issue with LaurentPolynomial because the following works as 
expected:

sage: Rq.<q> = PolynomialRing(ZZ,'q')
sage: Ruq.<u> = PolynomialRing(Rq,'u')
sage: mat = matrix([[q-u,u-q],[1,1]])
sage: mat.rref()
[1 0]
[0 1]



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