59.6 Ideals

For constructing and working with ideals in algebras the same functions are available as for ideals in rings. So for the precise description of these functions we refer to Chapter Rings. Here we give examples demonstrating the use of ideals in algebras. For an introduction into the concstruction of quotient algebras we refer to Chapter algebras of the user's tutorial.

gap> m:= [ [ 0, 2, 3 ], [ 0, 0, 4 ], [ 0, 0, 0] ];;
gap> A:= AlgebraWithOne( Rationals, [ m ] );;
gap> I:= Ideal( A, [ m ] );  # i.e., the two-sided ideal of `A' generated by `m'.  
<two-sided ideal in <algebra-with-one of dimension 3 over Rationals>, 
  (1 generators)>
gap> Dimension( I );
2
gap> GeneratorsOfIdeal( I );
[ [ [ 0, 2, 3 ], [ 0, 0, 4 ], [ 0, 0, 0 ] ] ]
gap> BasisVectors( Basis( I ) );
[ [ [ 0, 1, 3/2 ], [ 0, 0, 2 ], [ 0, 0, 0 ] ], 
  [ [ 0, 0, 1 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ]

gap> A:= FullMatrixAlgebra( Rationals, 4 );;
gap> m:= NullMat( 4, 4 );; m[1][4]:=1;;
gap> I:= LeftIdeal( A, [ m ] );
<left ideal in ( Rationals^[ 4, 4 ] ), (1 generators)>
gap> Dimension( I );
4
gap> GeneratorsOfLeftIdeal( I );
[ [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] ]

gap> mats:= [ [[1,0],[0,0]], [[0,1],[0,0]], [[0,0],[0,1]] ];;
gap> A:= Algebra( Rationals, mats );;
gap> I:= Ideal( A, [ mats[2] ], "basis" ); # we already know that the element
                                           # `mats[2]' forms a basis of the ideal.
<two-sided ideal in <algebra of dimension 3 over Rationals>, (dimension 1)>

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