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2.3.5 Modules and their annihilator

Now we shall give three more advanced examples.

SINGULAR is able to handle modules over all the rings, which can be defined as a basering. A free module of rank n is defined as follows:

ring rr;
int n = 4;
==> _[1]=gen(1)
==> _[2]=gen(2)
==> _[3]=gen(3)
==> _[4]=gen(4)
==> module
==> 1,0,0,0,
==> 0,1,0,0,
==> 0,0,1,0,
==> 0,0,0,1

To define a module, we provide a list of vectors generating a submodule of a free module. Then this set of vectors may be identified with the columns of a matrix. For that reason in SINGULAR matrices and modules may be interchanged. However, the representation is different (modules may be considered as sparse matrices).

ring r =0,(x,y,z),dp;
module MD = [x,0,x],[y,z,-y],[0,z,-2y];
matrix MM = MD;
==> x,y,0,
==> 0,z,z,
==> x,-y,-2y

However the submodule $MD$ may also be considered as the module of relations of the factor module $r^3/MD$.In this way, SINGULAR can treat arbitrary finitely generated modules over the basering (see Representation of mathematical objects).

In order to get the module of relations of $MD$, we use the command syz.

==> _[1]=x*gen(3)-x*gen(2)+y*gen(1)

We want to calculate, as an application, the annihilator of a given module. Let $M = r^3/U$,where U is our defining module of relations for the module $M$.

module U = [z3,xy2,x3],[yz2,1,xy5z+z3],[y2z,0,x3],[xyz+x2,y2,0],[xyz,x2y,1];

Then, by definition, the annihilator of M is the ideal $\hbox{ann}(M) = \{a \mid aM = 0 \}$which is, by definition of M, the same as $\{ a \mid ar^3 \in U \}$.Hence we have to calculate the quotient $U \colon r^3 $.The rank of the free module is determined by the choice of U and is the number of rows of the corresponding matrix. This may be retrieved by the function nrows. All we have to do now is the following:


The result is too big to be shown here.