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 (x²ⁿ)⁻ = 0 for some integer n ≥ 1 must be an O*-ring.

Mathematics Subject Classification (2010):

• Primary: 06F25

Key words:

• f-ring
• almost f-ring
• d-ring
• division closed
• power positive
• Σ-clean ring
• Σ-semiclean ring