Ideals of commutative Frobenius algebras

Question

Given a finite dimensional commutative (connected=local) Frobenius algebra $A$ over a field $K$.

Question 1: Does $A$ have only finitely many ideals? (the answer should be no in the non-commutative case, but Im not sure about the commutative case)

Question 2: In case the poset of ideals of two such algebras $A_1$ and $A_2$ are isomorphic, are $A_1$ and $A_2$ isomorphic in case they have the same vector space dimension?

Question 3:Is there a way to obtain the ideal lattice of a finite dimensional algebra via QPA? (Im not sure whether there is a good method to check whether one module is a submodule of another one (not up to isomorphism))

Answer 1

Question 1: Let $A=k[x,y]/\langle x^3, y^3\rangle$. A $k$-basis for $A$ is $\{1,x,y,x^2,xy,y^2,x^2y,xy^2, x^2y^2\}$. Let $I_q = \langle x^2y + qxy^2\rangle$, and a $k$-basis for $I_q$ is $\{ x^2y + qxy^2, x^2y^2\}$. Hence all the ideal $I_q$ are different, and if the field $k$ is infinite, there exists infinite many ideals.

Question 2: Don't know.

Question 3: For an infinite field the above example shows that there are infinitely many ideals, so I assume a finite field. Define the following algebra in QPA:

Q := Quiver( 1, [ [ 1, 1, "a" ], [ 1, 1, "b" ] ] );
KQ := PathAlgebra( GF(5), Q );
AssignGeneratorVariables( KQ );
relations := [ a^3, a * b - b * a, b^3 ];
A:= KQ / relations;
M := AlgebraAsModuleOverEnvelopingAlgebra( A );

Finding all the submodules of $A$ as a bimodules, that is, all ideals in $A$.

test := AllSubmodulesOfModule( M );

Say we want to check which $3$ dimensional modules the first $2$ dimensional module is contained in. Then we can do the following:

#
# Creating a vectorspace of the image of a homomorphism
#
f := test[ 3 ][ 1 ];
Imf := List( BasisVectors( Basis( Source( f ) ) ), m -> ImageElm( f, m )![ 1 ]![ 1 ] );
Imf := List( Imf, b -> Flat( b ) );
#
# Creating all the vectorspaces for the 3 dimensional submodules
#
dim3submodules := [ ];
for g in test[ 4 ] do
    Img := List( BasisVectors( Basis( Source( g ) ) ), m -> ImageElm( g, m )![ 1 ]![ 1 ] );
    Img := List( Img, b -> Flat( b ) );
    Add( dim3submodules, Subspace( GF(5)^9, Img ) );
od;
#
# Checking which ones contains Imf
#
List( dim3submodules, V -> ForAll( Imf, m -> m in V ) );

The output looks like this from the last line:

gap> List( dim3submodules, V -> ForAll( Imf, m -> m in V ) );
    [ true, true, true, true, true, true, false, false, false, false, false, false, false, false, false, false, false, false, false, 
      false, false, false, false, false, false, false, false, false, false, false, false ]

I hope that these comments are helpful.

The QPA-team.