Let
W
W
be a complete Noetherian local commutative ring with residue field
k
k
of positive characteristic
p
p
. We study the inverse problem for the universal deformation rings
R
W
(
Γ
,
V
)
R_{W}(\Gamma ,V)
relative to
W
W
of finite dimensional representations
V
V
of a profinite group
Γ
\Gamma
over
k
k
. We show that for all
p
p
and
n
≥
1
n \ge 1
, the ring
W
[
[
t
]
]
/
(
p
n
t
,
t
2
)
W[[t]]/(p^n t,t^2)
arises as a universal deformation ring. This ring is not a complete intersection if
p
n
W
≠
{
0
}
p^nW\neq \{0\}
, so we obtain an answer to a question of M. Flach in all characteristics. We also study the ‘inverse inverse problem’ for the ring
W
[
[
t
]
]
/
(
p
n
t
,
t
2
)
W[[t]]/(p^n t,t^2)
; this is to determine all pairs
(
Γ
,
V
)
(\Gamma , V)
such that
R
W
(
Γ
,
V
)
R_{W}(\Gamma ,V)
is isomorphic to this ring.