Semiprime f-Rings That Are Subdirect Products of Valuation Domains
Recall that an f-ring is a lattice-ordered ring in which a Λ b = 0 implies a Λ bc = a Λ cb = 0 whenever c ≥ 0. In [BKW], an f-ring is defined to be a lattice-ordered ring which is a subdirect product of totally ordered rings. These two definitions are equivalent if and only if the prime ideal theorem for Boolean Algebras is assumed; see [FH]. We regard these two definitions as equivalent henceforth. Our main concern is with f-rings that are semiprime; i.e., such that the intersection of the prime ideals is 0. A ring whose only nilpotent element is 0 is said to be reduced. (An f-ring is semiprime if and only if it is reduced; see [BKW, 8.5].) We will, however, maintain more generality when it does not take us too far afield. An ℓ-ideal I of an f-ring A is the kernel of a homomorphism of A into an f-ring. Equivalently, I is a ring ideal of A such that if a ∈ I, b ∈ A, and ∣b∣ < ∣a∣, then b ∈ I. A left ideal with this latter property is called a left ℓ-ideal, and a right ℓ-ideal is defined similarly. We let N(A) denote the set of nilpotent elements of the f-ring A.
© 1993 Kluwer Academic Publishers, published by Springer Netherlands
Henriksen, Melvin; Larson, Suzanne Semiprime f-rings that are subdirect products of valuation domains. Ordered algebraic structures (Gainesville, FL, 1991), 159–168, Kluwer Acad. Publ., Dordrecht, 1993. DOI: 10.1007/978-94-011-1723-4_10