Monogeny dimension relative to a fixed uniform module

For a module A and a uniform module U, we consider the invariant m-dim_U(A):=sup{i\in \N_0 | there exist morphisms f:U_i→A and g:A→U_i with gf a monomorphism}. This invariant turns out to have the following properties: (1) m-dim_U respects direct sums; (2) if U and V are uniform and [U]_m=[V]_m, then m-dim_U=m-dim_V; and (3) if two modules A and B have finite Goldie dimension and [A]_m=[B]_m, then m-dim_U(A)=m-dim_U(B) for every uniform module U. In particular, when A has finite Goldie dimension and is a direct summand of a serial module, the values m-dim_U(A) 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.