Throughout A denotes an f-ring; that is, a lattice-ordered ring that is a subdirect union of totally ordered rings. We let D(A) denote the set of derivations D: A --> A such that a ≥ 0 implies Da ≥ 0, and we call such derivations positive. In [CDK], P. Coleville, G. Davis, and K. Keimel initiated a study of positive derivations on f-rings. Their main results are (i) D ε D(A) and A archimedean imply D = 0, and (ii) if A has an identity element 1 and a is the supremum of a set of integral multiples of 1, then Da = O. Their proof of (i) relies heavily on the theory of positive orthomorphisms on archimedean f-rings and gives no insight into the general case. Below, in Theorem 4 and its corollary, we give a direct proof of (i), and in Theorem 10, we generalize (ii). Throughout, we improve on results in [CDK], and we study a variety of topics not considered therein.
© 1982 American Mathematical Society
Henriksen, Melvin, and F. A. Smith. "Some properties of positive derivations on f-rings." Proceedings of the Special Session on Ordered Field and Real Algebraic Geometry, 87th Annual Meeting of the American Mathematical Society (San Francisco, CA, 7-11 January 1981). Ed. D. W. Dubois and T. Recio. Contemporary Mathematics 8 (1982): 175–184.