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Let R be a ring with center C,
and ℐ be the additive group of all inner derivations of R. An additive group 𝒟 of
derivations of R is said to be a primary class of derivations of R if (i) for any ∂ ∈𝒟
and δ ∈ℐ, [∂,δ] ∈𝒟, (ii) for any x ∈ R, ∂x = 0 for all ∂ ∈𝒟 if and only if x ∈ C,
and (iii) for any prime ideal P in R and any x ∈ R, ∂x ∈ P for all ∂ ∈𝒟 if and only
if δx ∈ P for all δ ∈ℐ.
Suppose R has a primary class 𝒟 of derivations. First we assume, for each
x ∈ R and ∂ ∈𝒟, there is a p ∈ R such that ∂x − (∂x)2p ∈ C. Then all
nilpotent elements in R form an ideal N of R and R∕N is a subdirect sum of
division rings and commutative rings. If R is prime, then R has no nonzero
divisors of zero. Next, we assume that, for each x ∈ R and ∂ ∈𝒟, there is a
polynomial p(t) of t with integral coefficients such that ∂x − (∂x)2p(∂x) ∈ C or,
for each x ∈ R and ∂ ∈𝒟, there is a p ∈ C such that ∂x − (∂x)2p ∈ C.
Then ∂x ∈ C for all x ∈ R and ∂ ∈𝒟. If R is prime, then R is necessarily
commutative.
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