Vol. 80, No. 1, 1979

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Derivations and commutativity of rings

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

Vol. 80 (1979), No. 1, 77–89

Let R be a ring with center C and Jacobson radical J. Let be the additive group of all inner derivations of R and 𝒟 be an additive group of derivations of R satisfying:

  1. For any ∈𝒟 and δ ∈ℐ, [∂, δ] ∈𝒟;
  2. For any x R, ∂x = 0 for all ∈𝒟 iff x C;
  3. 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.

Mathematical Subject Classification
Primary: 16A72, 16A72
Secondary: 16A70
Received: 21 February 1978
Revised: 16 June 1978
Published: 1 January 1979
Lung O. Chung
Jiang Luh
Anthony N. Richoux