Local submodules of finite rings Let $A$ be a finite local ring $R$, such as a group algebra of $p$-groups over a finite field of characteristic $p$.

Question 1: Is it possible, using GAP, to obtain the poset of all submodules of $R$ of the form $uR$ up to equality where $u$ is an elements of $R$ (that are exactly the local submodules of $R$, that is the submodules with simple top)?
Question 2: Let $G_n$ be an elementary abelian $p$-group of order $p^n$ and $K$ the field with $p$ elements. What is the number of such submodules of the form $uR$ in this case when $R=KG$?

For $p=2$, the sequence starts with 2,5,23 for $n=1,2,3$. I wonder what the next term is?
 A: The following lines of code should work for a finite ring where all the commands used are defined (here there is an example of a group algebra):
G := SmallGroup( 8, 5 );
KG := GroupRing( GF(2), G );
rightideals := [ ];
rightideals := Unique( List( Elements( KG ), u -> RightIdeal( KG, [ u ] ) ) );

n := Length( rightideals );
relations := List( [ 1..n ], i -> [ i ] );

for i in [ 1..n - 1 ] do
    for j in [ i + 1..n ] do
        I := rightideals[ i ];
        J := rightideals[ j ];
        if Intersection( I, J ) = I then 
            Add( relations[ i ], j );
        elif Intersection( I, J ) = J then
            Add( relations[ j ], i );
        fi;
    od;
od;

_P := [ 1..n ];

D := Domain( _P );
_P := Elements( D );
rel := [];
for r in relations do
    for i in [ 2..Length( r ) ] do
        Add(rel, DirectProductElement( [ Elements( D )[Position( _P, r[ 1 ] ) ], Elements( D )[ Position( _P, r[ i ] ) ] ] ) );
    od;
od;
P := BinaryRelationByElements( D, rel );
P := ReflexiveClosureBinaryRelation( P );
P := TransitiveClosureBinaryRelation( P );
H := HasseDiagramBinaryRelation( P ); 
UnderlyingRelation( H );

However, it might take a while for big finite rings.
