Positivity 13 (2009), 165–191
2008 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010165-27, published online April 30, 2008
Positive derivations on archimedean
Jingjing Ma and Robert H. Redﬁeld
The second author dedicates this paper to J. B. Miller.
Abstract. It is known that the only positive derivation on a reduced
archimedean f-ring is the zero derivation. We investigate derivations on gen-
eral archimedean lattice-ordered rings. First, we consider semigroup rings over
cyclic semigroups and show that, in the ﬁnite case, the only derivation that is
zero on the underlying ring is the zero derivation and that, in the inﬁnite case,
such derivations are always based on the derivative. Turning our attention to
lattice-ordered rings, we show that, on many algebraic extensions of totally
ordered rings, the only positive derivation is the zero derivation and that, for
transcendental extensions, derivations that are lattice homomorphisms are
always translations of the usual derivative and derivations that are orthomor-
phisms are always dilations of the usual derivative. We also show that the
only positive derivation on a lattice-ordered matrix ring over a subﬁeld of the
real numbers is the zero derivation, and we prove a similar result for certain
lattice-ordered rings with positive squares.
Mathematics Subject Classiﬁcation (2000). Primary: 06F25; Secondary:
16W25, 13J25, 46G05.
Keywords. Derivation, positive derivation, archimedean lattice-ordered ring,
archimedean lattice-ordered algebra, polynomial ring, real matrix ring,
The set D(R) of real-valued diﬀerentiable functions of a real variable is an algebra
over R and the usual derivative
is a linear function on this algebra that does
not preserve multiplication but rather obeys the familiar product rule:
The second author thanks Hamilton College for its support of his visits to the ﬁrst author in
Houston. He also thanks John Miller for his friendship and hospitality over the last thirty years.