Monogeny dimension relative to a fixed uniform module

For a module A and a uniform module U, we consider the invariant image there exist morphisms f:Ui→A and g:A→Ui with gf a monomorphism}. This invariant turns out to have the following properties: (1) image for every image; (2) if U and V are uniform and [U]m=[V]m, then image; and (3) if image have finite Goldie dimension and [A]m=[B]m, then image for every uniform module U. In particular, when A has finite Goldie dimension and is a direct summand of a serial module, the values image completely determine the monogeny class of the module A. We give a complete description of the monoid of all isomorphism classes of serial modules of finite Goldie dimension over a fixed ring R.