Monogeny dimension relative to a fixed uniform module

For a module AA and a uniform module UU, we consider the invariant m-dimU(A)≔sup{i∈N0∣ there exist morphisms f:Ui→Af:Ui→A and g:A→Uig:A→Ui with gfgf a monomorphism}. This invariant turns out to have the following properties: (1) m-dimU(A⊕B)=m-dimU(A)+m-dimU(B) for every A,B∈Mod-R; (2) if UU and VV are uniform and [U]m=[V]m[U]m=[V]m, then m-dimU=m-dimV; and (3) if A,B∈Mod-R have finite Goldie dimension and [A]m=[B]m[A]m=[B]m, then m-dimU(A)=m-dimU(B) for every uniform module UU. In particular, when AA has finite Goldie dimension and is a direct summand of a serial module, the values m-dimU(A) completely determine the monogeny class of the module AA. We give a complete description of the monoid of all isomorphism classes of serial modules of finite Goldie dimension over a fixed ring RR.