Division closed partially ordered rings

Fuchs [6] called a partially-ordered integral domain, say D, division closed if it has the property that whenever a > 0 and ab > 0, then b > 0. He showed that if D is a lattice-ordered division closed field, then D is totally ordered. In fact, it is known that for a lattice-ordered division ring, the following three conditions are equivalent: a) squares are positive, b) the order is total, and c) the ring is division closed. In the present article, our aim is to study `-rings that possibly possess zerodivisors and focus on a natural generalization of the property of being division closed, what we call regular division closed. Our investigations lead us to the concept of a positive separating element in an `-ring, which is related to the well-known concept of a positive d-element. 1. Partially Ordered Rings Throughout, rings are assumed to be commutative. We do not assume that rings possess an identity, but when the ring R does possess an identity, we will denote it by 1R (or 1 when it is not ambiguous). A partially ordered ring (or po-ring for short) is a ring R equipped with a partial order, say ≤, such that the following two properties hold: (1) if a ≤ b, then a + c ≤ b + c, and (2) whenever 0 ≤ a, b then 0 ≤ ab. An equivalent, and more useful, way of viewing a partially ordered ring is through its positive cone: R = {a ∈ R : 0 ≤ a}. As is well-known, the positive cone of a po-ring can be characterized algebraically. Definition 1.1. Recall that a positive cone of a ring is a subset P ⊆ R such that the following three conditions are satisfied: i) P + P ⊆ P , ii) PP ⊆ P , and iii) P ∩ −P = {0}. Given a po-ring (R,≤), the set R is a positive cone. Conversely, a positive cone of R generates a partial order on R making R into a po-ring: define a ≤ b precisely when b − a ∈ P (see [5] or [13] for more information.) We shall use both notations (R,≤) and (R,R) to denote that R is a partially ordered ring. A partially ordered ring whose partial order is a lattice order is called a lattice-ordered ring (or `-ring for short). In particular, an `-ring is a latticeordered group and so the theory of such groups is useful in studying `-rings. If (R,R) is an `-ring, we denote the least upper bound (resp., greatest lower bound) of a, b ∈ R by a ∨ b (resp., a ∧ b). The positive part of a is a =