#### Title

Semiprime f-Rings That Are Subdirect Products of Valuation Domains

#### Document Type

Book Chapter

#### Department

Mathematics (HMC)

#### Publication Date

1993

#### Abstract

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*.

#### Rights Information

© 1993 Kluwer Academic Publishers, published by Springer Netherlands

#### DOI

10.1007/978-94-011-1723-4_10

#### Recommended Citation

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