For ideals in Sage, comparison (==) means mathematical equality. This is why 
there is both a class for ideals (MPolynomialIdeal) and for a sequence of 
polynomials in a common polynomial ring (PolynomialSequence). Yes it is 
expensive to test mathematical equality, so don't do it if you can't afford 
to wait. See the discussion 
at http://trac.sagemath.org/sage_trac/ticket/1819

I think it would be more consistent to demand mathematical equality for 
equality of the quotient rings. It is very likely that this is what the user 
actually wants if he/she compares two quotient rings.


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