Let R be a discrete valuation ring with maximal ideal
m
\mathfrak {m}
and the quotient field K. Let
Λ
=
(
m
λ
i
j
)
⊆
M
n
(
K
)
\Lambda = ({\mathfrak {m}^{{\lambda _{ij}}}}) \subseteq {M_n}(K)
be a tiled R-order, where
λ
i
j
∈
Z
{\lambda _{ij}} \in {\mathbf {Z}}
and
λ
i
i
=
0
{\lambda _{ii}} = 0
for
1
≤
i
≤
n
1 \leq i \leq n
. The following results are proved. Theorem 1. There are, up to conjugation, only finitely many tiled R-orders in
M
n
(
K
)
{M_n}(K)
of finite global dimension. Theorem 2. Tiled R-orders in
M
n
(
K
)
{M_n}(K)
of finite global dimension satisfy DCC. Theorem 3. Let
Λ
⊆
M
n
(
R
)
\Lambda \subseteq {M_n}(R)
and let
Γ
\Gamma
be obtained from
Λ
\Lambda
by replacing the entries above the main diagonal by arbitrary entries from R. If
Γ
\Gamma
is a ring and if gl
dim
Λ
>
∞
\dim \;\Lambda > \infty
, then gl
dim
Γ
>
∞
\dim \;\Gamma > \infty
. Theorem 4. Let
Λ
\Lambda
be a tiled R-order in
M
4
(
K
)
{M_4}(K)
. Then gl
dim
Λ
>
∞
\dim \;\Lambda > \infty
if and only if
Λ
\Lambda
is conjugate to a triangular tiled R-order of finite global dimension or is conjugate to the tiled R-order
Γ
=
(
m
λ
i
j
)
⊆
M
4
(
R
)
\Gamma = ({\mathfrak {m}^{{\lambda _{ij}}}}) \subseteq {M_4}(R)
, where
γ
i
i
=
γ
1
i
=
0
{\gamma _{ii}} = {\gamma _{1i}} = 0
for all i, and
γ
i
j
=
1
{\gamma _{ij}} = 1
otherwise.