Let F be a mixed free algebra on a set X over the field K. Let U, V be two ideals of F, and
{
δ
(
x
)
,
(
x
∈
X
)
}
\{ \delta (x),(x \in X)\}
a basis for a free
(
F
/
U
,
F
/
V
)
(F/U,F/V)
-bimodule T. Then the map
x
→
(
x
+
V
a
m
p
;
0
δ
(
x
)
a
m
p
;
x
+
U
)
x \to (\begin {array}{*{20}{c}} {x + V} & 0 \\ {\delta (x)} & {x + U} \\ \end {array} )
induces an injective homomorphism
F
/
U
V
→
(
F
/
V
a
m
p
;
0
T
a
m
p
;
F
/
U
)
F/UV \to (\begin {array}{*{20}{c}} {F/V} & 0 \\ T & {F/U} \\ \end {array} )
. If
F
/
U
F/U
and
F
/
V
F/V
are embeddable in matrices over a commutative algebra, so is
F
/
U
V
F/UV
. Some special cases are investigated and it is shown that a PI algebra with nilpotent radical satisfies all identities of some full matrix algebra.