Positive derivations on Archimedean lattice-ordered rings
type: Item
instance of: scholarly article
title: Positive derivations on Archimedean lattice-ordered rings
zbMATH Open document ID: 1167.06012
author: Jingjing Ma
author: R. H. Redfield
published in: Positivity
publication date: 2009-03-27T00:00:00Z
review text: 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. [\textit{P. Colville, G. Davis} and \textit{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.
reviewed by: Igor Klep
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 13J25
Mathematics Subject Classification ID: 16W25
Mathematics Subject Classification ID: 46G05
zbMATH DE Number: 5534423
zbMATH Keywords: derivative
zbMATH Keywords: positive derivation
zbMATH Keywords: Archimedean lattice-ordered ring
zbMATH Keywords: polynomial ring
zbMATH Keywords: matrix ring
zbMATH Keywords: positive squares
MaRDI profile type: Publication
full work available at URL: s11117-008-2183-1
OpenAlex ID: W2012789285
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DOI: 10.1007/S11117-008-2183-1
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description: scientific article; zbMATH DE number 5534423