Singular

Hello,

A question related not directly with Singular, but with algebraic geometry:
The geometric interpretation of a syzygy module is that it forms the tangent bundle of the algebraic set of the corresponding ideal, respectively module.
When now one of the elements m_i from the basis of the module is multiplied with a polynomial f, it is still in the module, however the module generated by the new generating set is smaller. I was wondering what the geometric interpretation of this is, e.g. does the new module describe the tangent space of the ideal everywhere where f=!=0?
This problem occured when I was trying to solve a huge polynomial equation and took the quick and dirty path of using Cramers rule, which gives a solution which is complete in the sense of algebra, but not in the sense of algebraic geometry, so I was wondering what these solutions describe geometrically.

Best regards and Thanks in advance!