In this paper, we prove the following two results which generalize the theorem concerning automorphic-differential endomorphisms asserted by J. Bergen. Let
R
R
be a ring,
R
F
R _{\mathcal {F}}
its left Martindale quotient ring and
A
\mathfrak {A}
a right ideal of
R
R
having no nonzero left annihilator. (1) Let
C
C
be a pointed coalgebra which measures
R
R
such that the group-like elements of
C
C
act as automorphisms of
R
R
. If
R
R
is prime and
ξ
⋅
A
=
0
\xi \cdot \mathfrak {A}=0
for
ξ
∈
R
#
C
\xi \in R\#C
, then
ξ
⋅
R
=
0
\xi \cdot R=0
. Furthermore, if the action of
C
C
extends to
R
F
R _{\mathcal {F}}
and if
ξ
∈
R
F
#
C
\xi \in R _{\mathcal {F}}\#C
such that
ξ
⋅
A
=
0
\xi \cdot \mathfrak {A}=0
, then
ξ
⋅
R
F
=
0
\xi \cdot R _{\mathcal {F}}=0
. (2) Let
f
f
be an endomorphism of
R
F
R _{\mathcal {F}}
given as a sum of composition maps of left multiplications, right multiplications, automorphisms and skew-derivations. If
R
R
is semiprime and
f
(
A
)
=
0
f(\mathfrak {A})=0
, then
f
(
R
)
=
0
f(R)=0
.