Positive derivations on lattice ordered rings of matrices

Abstract

Here we consider a commutative ℓ-ring R with unit. We show that the only positive derivation on the ℓ-ring M_n (R) is 0 if and only if the only positive derivation on R is zero. Also, if the only positive derivation on R is zero, we prove that for certain semiprime sub ℓ-rings S of M_n (R) the only positive derivation on S is zero. Then, we study positive derivations on sub ℓ-rings of M_n ( ^R ) where R is an Archimedean ℓ-ring.