Let
R
R
be any associative ring. Suppose that for every pair
(
a
1
,
a
2
)
∈
R
×
R
({a_1},{a_2}) \in R \times R
there exists a pair
(
p
1
,
p
2
)
({p_1},{p_2})
such that the elements
a
i
−
a
i
2
p
i
(
a
i
)
{a_i} - a_i^2{p_i}({a_i})
commute, where the
p
i
{p_i}
’s are polynomials over the integers with one (central) indeterminate. It is shown here that the nilpotent elements of
R
R
form a commutative ideal
N
N
, and that the factor ring
R
/
N
R/N
is commutative. This result is obtained by the use of the concept of cohypercenter of a ring
R
R
, which concept parallels the hypercenter of a ring.