Let
R
=
R
1
∐
R
2
R = {R_1}\coprod {R_2}
be the coproduct of
Δ
\Delta
-rings
R
1
{R_1}
and
R
2
{R_2}
with 1 over a division ring
Δ
,
R
1
≠
Δ
,
R
2
≠
Δ
\Delta ,\qquad {R_1} \ne \Delta ,\qquad {R_2} \ne \Delta
, with at least one of the dimensions
(
R
i
:
Δ
)
r
,
(
R
i
:
Δ
)
l
,
i
=
1
,
2
{({R_i}:\Delta )_r},\,{({R_i}:\Delta )_l},\,i = 1,\,2
, greater than 2. If
R
1
{R_1}
and
R
2
{R_2}
are weakly
1
1
-finite (i.e., one-sided inverses are two-sided) then it is shown that every
X
X
-inner automorphism of
R
R
(in the sense of Kharchenko) is inner, unless
R
1
,
R
2
{R_1},\,{R_2}
satisfy one of the following conditions: (I) each
R
i
{R_i}
is primary (i.e.,
R
i
=
Δ
+
T
,
T
2
=
0
{R_i} = \Delta + T,\,{T^2} = 0
), (II) one
R
i
{R_i}
is primary and the other is
2
2
-dimensional, (III) char.
Δ
=
2
\Delta = 2
, one
R
i
{R_i}
is not a domain, and one
R
i
{R_i}
is
2
2
-dimensional. This generalizes a recent joint result with Lichtman (where each
R
i
{R_i}
was a domain) and an earlier joint result with Montgomery (where each
R
i
{R_i}
was a domain and
Δ
\Delta
was a field).