Let
C
{\mathbf {C}}
be an additive category such that idempotent endomorphisms have kernels,
C
C
a class of objects of
C
{\mathbf {C}}
having Dedekind domains as endomorphism rings, and assume that if
X
X
and
Y
Y
are quasi-isomorphic objects of
C
C
then
Hom
(
X
,
Y
)
{\operatorname {Hom}}(X,Y)
is a torsion-free module over the endomorphism ring of
X
X
.
A
⊕
B
=
C
1
⊕
⋯
⊕
C
n
A \oplus B = {C_1} \oplus \cdots \oplus {C_n}
with each
C
i
{C_i}
in
C
C
, then
A
=
A
1
⊕
⋯
⊕
A
m
A = {A_1} \oplus \cdots \oplus {A_m}
, where each
A
j
{A_j}
is locally in
C
C
, and
End
(
A
j
)
≃
End
(
C
i
)
{\operatorname {End}}({A_j}) \simeq {\operatorname {End}}({C_i})
for some
i
i
. The proof includes a characterization of tiled orders. Moreover, there is a "local" uniqueness for finite direct sums of objects of
C
C
.