Let
R
R
be a commutative ring. In (1), it was proved that a ring
R
R
with noetherian total quotient ring is self-injective if and only if the endomorphism ring of every ideal is commutative. We prove here that if the ring is coherent and is its own total quotient ring, then
R
R
is self-injective if and only if
Hom
(
I
,
I
)
=
R
\operatorname {Hom} (I,I) = R
for every ideal
I
I
of
R
R
.