Let G be a module over a ring R, let
C
=
{
C
i
}
,
i
∈
I
\mathcal {C} = \{ {C_i}\}, i \in I
, be a family of submodules of G, and let
H
=
{
H
i
}
,
i
∈
I
\mathcal {H} = \{ {H_i}\}, i \in I
, where
H
i
{H_i}
is a subgroup of
Hom
R
(
C
i
,
G
)
\operatorname {Hom}_R({C_i},G)
with certain properties. To each such pair
(
C
,
H
)
(\mathcal {C},\mathcal {H})
, a near-ring
M
(
C
,
H
)
M(\mathcal {C},\mathcal {H})
is associated, which is a generalization of the near-ring of homogeneous functions determined by (G, R). The transfer of information from module properties of
G
R
{G_R}
reflected in
(
C
,
H
)
(\mathcal {C},\mathcal {H})
to structural properties of
M
(
C
,
H
)
M(\mathcal {C},\mathcal {H})
is investigated.