Let R be a ring with center C
and Jacobson radical J. Let ℐ be the additive group of all inner derivations of R and
𝒟 be an additive group of derivations of R satisfying:
For any ∂ ∈𝒟 and δ ∈ℐ, [∂, δ] ∈𝒟;
For any x ∈ R, ∂x = 0 for all ∂ ∈𝒟 iff x ∈ C;
For any prime ideal P in R and any x ∈ R, ∂x ∈ P for all ∂ ∈𝒟 iff δx ∈ P
for all δ ∈ℐ.
Suppose, for each x ∈ R and ∂ ∈𝒟, there is a p ∈ R which depends upon x and ∂
such that ∂x = (∂x)2p. Then the nilpotent elements in R are central and form an
ideal N in R, R∕N is a subdirect sum of division rings and commutative rings, and
R∕J is a subdirect sum of division rings. Suppose further that, for each x ∈ R and
∂ ∈𝒟, such a p is a polynomial of ∂x with integral coefficients. Then R is necessarily
commutative.
