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2.3.5 Modules and their annihilatorNow 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
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).
However the submodule 10#10 may also be considered as the module of relations of the factor module 11#11.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
10#10,
we use the command
We want to calculate, as an application, the annihilator of a given module. Let 12#12,where U is our defining module of relations for the module 13#13.
Then, by definition, the annihilator of M is the ideal
14#14which is, by definition of M, the same as
15#15.Hence we have to calculate the quotient
16#16.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
The result is too big to be shown here. |
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