On Tuesday, June 12, 2018 at 2:09:53 PM UTC-7, Simon King wrote:
> Problem: 
> It is currently possible to create the polynomial ring R[x][x] 
> (hence, x occurs twice). If "x" has two different meanings in the 
> same ring, the notion "name preserving map" makes no sense. I believe 
> it should be made impossible to create such ring. Perhaps the .__mul__ 
> method of PolynomialRing and MultiPolynomialRing could be utilised for 
> this, when needed. 
There is a very good reason to allow such rings to exist:

sage: matrix(QQ['y'],3,3,1).characteristic_polynomial()
x^3 - 3*x^2 + 3*x - 1
sage: matrix(QQ['x'],3,3,1).characteristic_polynomial()
x^3 - 3*x^2 + 3*x - 1

If you forbid them, then characteristic_polynomial needs to start working 
very hard to come up with a sensible ring.

Of course, any name-based coercion discovery is allowed to fail horribly 
with the results, but I think we need to keep them around as place holder 
for these automatically constructed parents.

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