Division closed ℓ-rings and power positive L^∗ -rings

Abstract

A commutative non-associative division closed lattice-ordered ring with identity that is not an f -ring is presented. More conditions are provided to ensure that an associative division closed lattice-ordered ring is an f -ring. In particular, for a division closed lattice-ordered ring with identity, if it is Σ-clean or Σ-semiclean, then it is an f -ring. Finally it is shown that a ring with identity in which each partial order can be extended to a lattice order satisfying (x ²ⁿ )⁻ = 0 for some integer n ≥ 1 must be an O*-ring.