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Semiprime f-Rings That Are Subdirect Products of Valuation Domains

In: Ordered Algebraic Structures

Author

Listed:
  • Melvin Henriksen

    (Harvey Mudd College)

  • Suzanne Larson

    (Loyola Marymount University)

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∣

Suggested Citation

  • Melvin Henriksen & Suzanne Larson, 1993. "Semiprime f-Rings That Are Subdirect Products of Valuation Domains," Springer Books, in: J. Martinez & C. Holland (ed.), Ordered Algebraic Structures, pages 159-168, Springer.
    • Handle: RePEc:spr:sprchp:978-94-011-1723-4_10
    DOI: 10.1007/978-94-011-1723-4_10
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