Invariance of minimal prime ideals under derivations

Abstract

We construct a counterexample to refute the following conjecture by I. N. Herstein: Is every minimal prime ideal of a semiprime ring invariant under any derivation of this ring? The following conjecture was raised by I. N. Herstein and, recently, has been often mentioned in the literature. See, for example, [4, 8, 9]. Conjecture. If P is a minimal prime ideal of a semiprime ring R, is 3(P) C P for every derivation 6 of R ? The primary objective of this note is to refute the conjecture in general. However, the motivation behind this conjecture is to be able to reduce questions regarding derivations in semiprime rings to questions about prime rings and thus the real concern about a given semiprime ring is whether the intersection of the minimal prime ideals invariant under a given derivation is 0 or not. This problem is also due to I. N. Herstein and was listed as the second open question in [9]. In the counterexample constructed here, there is only one miniInal prime ideal that is not invariant under a derivation and the intersection of all the minimal prime ideals invariant under this given derivation is still 0. Thus it remains open as to whether the intersection of all the minimal prime ideals invariant under a given derivation is always 0 or not. Before proceeding to the construction of our counterexample, it might be interesting to review some positive partial results of the conjecture. There are mainly two directions; one is by restricting the characteristic and the other is by restricting the nilpotency index. For the direction of restricting the characteristic, Krempa [7] proved that any minimal prime ideals of a semiprime algebra over a field of characteristic 0 are always invariant under derivations. Actually, a much better result in this direction is contained in Propositions 1.1 and 1.3 of [5]. Specifically, the following are shown in [5]: (1) If P is a minimal prime ideal of a ring R such that R/P has characteristic 0, then P is invariant under any derivations of R. Received by the editors January 5, 1990 and, in revised form, August 23, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A72, 16A12.