Constructing a quotient ring in GAP using structure constants

Question

I need to construct the following ring in GAP:

$$Z_4(u) / \langle u^2-2u=0 \rangle. $$

This is what I tried and it didn't work:

gap> R:=PolynomialRing(Integers mod 4,"u");AssignGeneratorVariables(R);
<monoid>[u]
#I  Global variable `u' is already defined and will be overwritten
#I  Assigned the global variables [ u ]
gap> I:=Ideal(R,[u^2-2*u]);
<two-sided ideal in <monoid>[u], (1 generators)>
gap> A:=R/I;
Error, no method found! For debugging hints type ?Recovery from        
NoMethodFound
Error, no 3rd choice method found for `NaturalHomomorphismByIdeal' on 2
arguments called from NaturalHomomorphismByIdeal( A, I ) called from
<function "unknown">( <arguments> )
called from read-eval loop at line 15 of *stdin*
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk>

Does anyone know how to construct this ring in GAP?

Thanking you in advance!

Answer 1

The ideal quotient algorithms are only implemented for polynomial rings over fields. The only way to construct this ring would be by structure constants.

For example here you would have two generators ($a$ and $u$). The structure constant table encodes the multiplication of the generators

gap> T:=EmptySCTable(2,0);; # generators, zero coefficient
gap> SetEntrySCTable(T,1,1,[1,1]); # a*a=1*a (list is [coeff,gen,....
gap> SetEntrySCTable(T,1,2,[1,2]); # a*u=1*u
gap> SetEntrySCTable(T,2,1,[1,2]); # u*a=1*u
gap> SetEntrySCTable(T,2,2,[2,2]); # u*u=2*u

Both generators have additive order 4, and we want them to be printed as a and u

gap> r:=RingByStructureConstants([4,4],T,["a","u"]);
<ring with 2 generators>
gap> Elements(r);
[ 0*a, u, 2*u, -u, a, a+u, a+2*u, a-u, 2*a, 2*a+u, 2*a+2*u, 2*a-u, -a, -a+u, 
  -a+2*u, -a-u ]