Quotient of Polynomial Ring (Leading Coefficient)

edit

You could try this:

sage: from sage.rings.quotient_ring import QuotientRing_generic
sage: R.<x> = Integers(4)[]
sage: I = R.ideal(2*x)
sage: Q = QuotientRing_generic(R, I, ['a'])
sage: Q.inject_variables()

but I wouldn't be surprised if some things were broken. Quotients of polynomial rings are well implemented only when the leading coefficient is a unit, as the error message implies.

Edit: this doesn't raise any errors when you define the ring:

sage: R.<x> = ZZ[]
sage: J = R.ideal(2*x, 4)
sage: Q = R.quotient(J)

Maybe you can get somewhere this way.

EDIT : this answer is FALSE : see John Palmieri's comment. I leave it for the edification of future ask.sagemath.org (per-)users.

You can also explicitely use the fact that the ring of congruences modulo 4 is a field :

sage: R.<x>=Zmod(4)[]
sage: Q.<a>=R.quotient(2*x)
---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)

[ Snip... ]

TypeError: polynomial must have unit leading coefficient

But :

sage: R.<x>=GF(4)[]
sage: Q.<a>=R.quotient(2*x)
sage: Q
Univariate Polynomial Ring in x over Finite Field in z2 of size 2^2

HTH,