Let
R
R
be a ring and
f
f
an endomorphism obtained from sums and compositions of left multiplications, right multiplications, automorphisms, and derivations. We prove several results relating the behavior of
f
f
on certain subsets of
R
R
to its behavior on all of
R
R
. For example, we prove (1) if
R
R
is prime with ideal
I
≠
0
I \ne 0
such that
f
(
I
)
=
0
f(I) = 0
, then
f
(
R
)
=
0
f(R) = 0
, (2) if
R
R
is a domain with right ideal
λ
≠
0
\lambda \ne 0
such that
f
(
λ
)
=
0
f(\lambda ) = 0
, then
f
(
R
)
=
0
f(R) = 0
, and (3) if
R
R
is prime and
f
(
λ
n
)
=
0
f({\lambda ^n}) = 0
, for
λ
\lambda
a right ideal and
n
≥
1
n \geq 1
, then
f
(
λ
)
=
0
f(\lambda ) = 0
. We also prove some generalizations of these results for semiprime rings and rings with no non-zero nilpotent elements.