The Lie structure of a commutative ring with a derivation

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Ro], for all n > 0. Let A be a non-zero Lie ideal of R o and assume that R is 2-torsion free. In this situation C.R. Jordan and D.A. Jordan ([3], [4]) proved the following four theorems: (1) IfR is prime and A = R o or A = R 1 then A is a prime Lie ring ([3] Theorem 6, [4] Theorems I and 4). (2) If R is noetherian d-prime and A = R o or A = R 1 then A is a prime Lie ring ([3] Theorem 7, [4] Theorems 2 and 5). (3) If R is noetherian d-prime and A = R o or A = R 1 then every non-zero Lie ideal of A contains a non-zero d-ideal of R ([3] Theorem 8, [4] Theorems 3 and 6). [4] If R is noetherian then R o is simple if and only if R is d-simple ([4] Theorem 3). In this paper we show that Theorems (1)-(3) are also true in the case where A =