Quotient Rings of a Class of Lattice-Ordered-Rings

Abstract

An f-ring R with zero right annihilator is called a qf-ring if its Utumi maximal left quotient ring Q = Q(R) can be made into and f-ring extension of R. F. W. Anderson [2, Theorem 3.1] has characterized unital qf-rings with the following conditions: For each q ∈ Q and for each pair d1, d2 ∈ R+ such that diq ∈ R(i) (d1q)+ Λ (d2q)- = 0, and(ii) d1 Λ d2 = 0 implies (d1q)+ Λ d2 = 0.We remark that this characterization holds even when R does not have an identity element.

Rings of quotients of rings without nilpotent elements

“…In [20,Theorem 3.5] we have shown that if R is an /-ring in which each right ideal that has a finite number of positive generators has a single positive generator, then Q is an /-ring extension of R. This has motivated the next result. Proof.…”

“…To give a concrete example let R be the two-by-two matrix ring over the rationale and let A -e n R be the right ideal of R consisting of those matrices whose second row is zero. Then A is a right injective ring that is clearly not regular [20,Example 5.3].…”

A characterization of rings in which each partial order is contained in a total order

“…Suppose that R is of type (iii) and that P is a partial order of R. If x = a b 0 a ∈ P with a < 0 then we may assume that a = −1. But then −1 = It is interesting to note that the unique totally ordered (right or left) self-injective rings that are not unital are O * -rings [8,Theorem 5.4].…”

Lattice-ordered modules of quotients

“…An /-ring is called right convex if each of its right ideals is aconvex /-subgroup. Regular /-rings and left injective/-rings are right convex (Steinberg (1973)). Georgoudis (1972) has shown that each/-module is a .9/-module ifR is an / : ring that is either commutative or right convex.…”

“…Georgoudis (1972) has shown that each/-module is a .9/-module ifR is an / : ring that is either commutative or right convex. Also, see Anderson (1965) and Steinberg (1973). Henriksen(1977), p. 407, has considered the following condition on an /-ring R. Note that all commutative po-rings, right convex /-rings and left /-rings satisfying Henriksen's condition are (**)-rings.…”