The main result of this paper
states that for a lattice-ordered ring (l-ring) A with no nonzero nilpotent l-ideals the
following are equivalent: (i) A is an f-ring; (ii) A is a subdirect union of
totally-ordered rings with no nonzero divisors of zero; (iii) x+x−= 0 for all x ∈ A;
(iv) x+ax−= 0 for all x,a ∈ A; and (v) a(b ∨ c) = ab ∨ ac and (b ∨ c)a = ba ∨ ca for
all a,b,c ∈ A with a ≧ 0. In particular, the equivalence of (i) and (iii) implies that an
l-ring which has an identity that is a weak order unit and which has no nonzero
nilpotent l-ideals is necessarily an f-ring.
The basic tool in our considerations is the notion of prime l-ideal. Specifically, call
a proper l-ideal P of an l-ring A prime if I ⊆ P or J ⊆ P wherever I and J are
l-ideals of A with IJ ⊆ P. Various conditions are obtained on A, each of which forces
A modulo every prime l-ideal to be totally-ordered with no nonzero divisors of zero.
Moreover the relationship between the join of all the nilpotent l-ideals of A and the
intersection of all the prime l-ideals of A is investigated in order to obtain the
theorem mentioned above.
