In this paper, we construct a counter example to a conjecture of Johns to the effect that a right Noetherian ring in which every right ideal is an annihilator is right Artinian. Our example requires the existence of a right Noetherian domain
A
A
(not a field) with a unique simple right module
W
W
such that
W
A
{W_A}
is injective and
A
A
embeds in the endomorphism ring
End
(
W
A
)
\operatorname {End} ({W_A})
. Then the counter example is the trivial extension
R
=
A
⋉
W
R = A \ltimes W
of
A
A
and
W
W
. The ring
A
A
exists by a theorem of Resco using a theorem of Cohn. Specifically, if
D
D
is any countable existentially closed field with center
k
k
, then the right and left principal ideal domain defined by
A
=
D
⊗
k
k
(
x
)
A = D{ \otimes _k}k(x)
, where
k
(
x
)
k(x)
is the field of rational functions, has the desired properties, with
W
A
≈
D
A
{W_A} \approx {D_A}
.