Positive derivations on Archimedean lattice-ordered rings. (English) Zbl 1167.06012

Ma, Jingjing; Redfield, Robert H.

Positivity 13, No. 1, 165-191 (2009).

A lattice-ordered ring \(A\) is called an \(f\)-ring if \(x\land y=0\) and \(0\leq z\) imply \(zx\land y=xz\land y=0\) for all \(x,y,z\in A\). It is well-known that the only positive derivation (i.e., one that maps positive elements to positive elements) on a reduced Archimedean \(f\)-ring is the zero derivation, cf. [P. Colville, G. Davis and K. Keimel, J. Aust. Math. Soc., Ser. A 23, 371–375 (1977; Zbl 0376.06021)].
The paper under review examines the situation for general Archimedean lattice-ordered rings. The beginning sections focus on derivatives and derivations on certain polynomial and group rings. For instance, it is shown that for group rings of finite cyclic groups the only derivation vanishing on the underlying ring is the zero derivation. For infinite cyclic groups such derivations are always based on the derivative, see Corollary 3.4 or Theorem 3.2 for a more general statement.
In the second half of the article the authors turn their attention to derivatives and derivations on Archimedean lattice-ordered rings. It is proved that on purely transcendental extensions of totally ordered fields derivations that are at the same time lattice homomorphisms, are translations of the usual derivative (Theorem 9.1), and on algebraic extensions of totally ordered fields often the only positive derivation is the zero derivation (see Section 6 for detailed statements). The authors also give some results for lattice-ordered matrix rings (Section 8) and lattice-ordered rings in which all squares are positive (Section 7).
The paper abounds with examples that illustrate the results and show the necessity of hypotheses.

Reviewer: Igor Klep (Ljubljana)

MSC:

06F25	Ordered rings, algebras, modules
13J25	Ordered rings
16W25	Derivations, actions of Lie algebras
46G05	Derivatives of functions in infinite-dimensional spaces

Keywords:

derivative; positive derivation; Archimedean lattice-ordered ring; polynomial ring; matrix ring; positive squares

Citations:

Zbl 0376.06021

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