Ideals of $F_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2)$ I am interested in the poset of all ideals of the local ring
$$R_n = \mathbb{F}_2[x_1, x_2, \cdots, x_n]/(x_1^2, x_2^2, \cdots x_n^2).$$
$n=1$ is trivial. $n=2$ takes little work and it is shown below.

But beyond that, it is getting tedious. Is there a description of this lattice of ideals in $R_n$?  Or can this be computed using some software?
I am actually interested in computing all quotients of $R_n$.
Any help would be appreciated. Thank you!
 A: The ideals correspond to submodules of the regular module and as this is a finite dimensional commutative quiver algebra, you can obtain all submodules using the GAP package QPA (at least for finite fields).
Example for $n=2$ over a finite field with $2^t$ elements for $t \geq 1$:
t:=1;;K:=GF(2^t);Q:=Quiver(1,[[1,1,"x1"],[1,1,"x2"]]);KQ:=PathAlgebra(K,Q);AssignGeneratorVariables(KQ);rel:=[x1^2,x2^2,x1*x2-x2*x1];A:=KQ/rel;RegA:=IndecProjectiveModules(A)[1];W:=AllSubmodulesOfModule(RegA);WW:=Flat(W);Size(WW);

The output are the ideals with given as inclusions into the regular module, here:
[ [ <<[ 0 ]> ---> <[ 4 ]>> ], [ <<[ 1 ]> ---> <[ 4 ]>> ], [ <<[ 2 ]> ---> <[ 4 ]>>, <<[ 2 ]> ---> <[ 4 ]>>, <<[ 2 ]> ---> <[ 4 ]>> ], [ <<[ 3 ]> ---> <[ 4 ]>> ], [ <<[ 4 ]> ---> <[ 4 ]>> ] ],[ <<[ 0 ]> ---> <[ 4 ]>>, <<[ 1 ]> ---> <[ 4 ]>>, <<[ 2 ]> ---> <[ 4 ]>>, <<[ 2 ]> ---> <[ 4 ]>>, <<[ 2 ]> ---> <[ 4 ]>>, <<[ 3 ]> ---> <[ 4 ]>>, <<[ 4 ]> ---> <[ 4 ]>> ]

The number of ideals for a finite field with $2^t$ elements and $n=2$ starts for $t \geq 1$ with 7, 9, 13, 21, 37, 69, 133 and thus seems to be given by https://oeis.org/A168614 and depends strongly on the field (in general it seems the number of ideals for a finite field with $p^t$ elements is given by $p^t+5$).
The number of ideals increases with the number of elements of the finite field. Thus there surely will be infinitely many ideals (even up to isomorphism) for an infinite field, for example over a field of characteristic zero the ideals $(x_1+x_2)$ and $(q x_1 + x_2)$ are not isomorphic for $q \neq 1$.
You can also compute all quotients with QPA, having the inclusion maps.
