#### Vol. 85, No. 1, 1979

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Derivations and commutativity of rings. II

### Lung O. Chung, Jiang Luh and Anthony N. Richoux

Vol. 85 (1979), No. 1, 19–34
##### Abstract

Let R be a ring with center C, and 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.

##### Mathematical Subject Classification 2000
Primary: 16A70, 16A70
Secondary: 17A35, 16A72
##### Milestones 