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I think I found a workaround: make x not be the variable of the polynomial ring but rather that of the quotient of that ring by the ideal generated by z^2-z. In other words:

var('X'); x=PolynomialRing(Integers(3), 'X').quotient([X^2-1], 'x').gens()[0]
print x^3==x and x^2==1 and 3*x==0

I think I found a workaround: make x not be the variable of the polynomial ring but rather that of the quotient of that ring by the ideal generated by z^2-z. In other words:

var('X'); x=PolynomialRing(Integers(3), 'X').quotient([X^2-1], 'x').gens()[0]
print x^3==x and x^2==1 and 3*x==0