Let
A
,
B
A,\;B
be artinian rings and let
A
M
B
_A{M_B}
be an
(
A
−
B
)
(A - B)
-bimodule which is a finitely generated left
A
A
-module and a finitely generated right
B
B
-module. A right
A
M
B
_A{M_B}
-prinjective module is a finitely generated module
X
R
=
(
X
A
′
,
X
B
,
φ
:
X
A
′
⊗
A
M
B
→
X
B
)
{X_R} = (X_A’, X_B, \varphi :X_A’ \otimes _A M_B \to X_B)
over the triangular matrix ring
\[
R
=
(
A
a
m
p
;
A
M
B
0
a
m
p
;
B
)
R = \left ( {\begin {array}{*{20}{c}} A & {_A{M_B}} \\ 0 & B \\ \end {array} } \right )
\]
such that
X
A
′
X_A’
is a projective
A
A
-module,
X
B
X_B
is an injective
B
B
-module, and
φ
\varphi
is a
B
B
-homomorphism. We study the category
prin
(
R
)
B
A
\operatorname {prin} (R)_B^A
of right
A
M
B
_A{M_B}
-prinjective modules. It is an additive Krull-Schmidt subcategory of
mod
(
R
)
\bmod (R)
closed under extensions. For every
X
,
Y
X,\;Y
in
prin
(
R
)
B
A
,
Ext
R
2
(
X
,
Y
)
=
0
\operatorname {prin} (R)_B^A,\;\operatorname {Ext} _R^2(X,\,Y) = 0
. When
R
R
is an Artin algebra, the category
prin
(
R
)
B
A
\operatorname {prin} (R)_B^A
has Auslander-Reiten sequences and they can be computed in terms of reflection functors. In the case that
R
R
is an algebra over an algebraically closed field we give conditions for
prin
(
R
)
B
A
\operatorname {prin} (R)_B^A
to be representation-finite or representation-tame in terms of a Tits form. In some cases we calculate the coordinates of the Auslander-Reiten translation of a module using a Coxeter linear transformation.