Let
R
R
be a ring and
A
(
R
)
=
{
x
∈
R
:
x
A(R) = \{ x \in R:x
belongs to the second commutant of
{
x
n
,
x
n
+
1
}
\{ {x^n},{x^{n + 1}}\}
for all integers
n
>
1
}
n > 1\}
. It is shown that in a prime ring
R
,
A
(
R
)
=
R
R,A(R) = R
if and only if
R
R
has no nilpotent elements. The set
A
(
U
)
A(U)
is studied for some special
∗
\ast
-algebras. It is shown that the normal elements of a proper
∗
\ast
-algebra
U
U
belong to
A
(
U
)
A(U)
. If
U
U
is also prime then
A
(
U
)
=
{
x
∈
U
:
x
A(U) = \{ x \in U:x
belongs to the second commutant of
{
x
n
,
x
n
+
1
}
\{ {x^n},{x^{n + 1}}\}
for some
n
>
1
}
n > 1\}
. The set
A
(
B
(
H
)
)
A(B(H))
is studied, where
B
(
H
)
B(H)
is the algebra of bounded operators on a Hilbert space
H
H
. Necessary and sufficient conditions for some special types of operators to belong to
A
(
B
(
H
)
)
A(B(H))
are obtained.