Semiprime f-Rings That Are Subdirect Products of Valuation Domains

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Book Chapter


Mathematics (HMC)

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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 -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 aI, bA, and ∣b∣ < ∣a∣, then bI. A left ideal with this latter property is called a left -ideal, and a right -ideal is defined similarly. We let N(A) denote the set of nilpotent elements of the f-ring A.

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© 1993 Kluwer Academic Publishers, published by Springer Netherlands