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- W318703076 abstract "Abstract. Let R be a commutative ring with identity, and letf, g i (i = 1,...,n), g α (α ∈ S) be elements of R. We show thatthe following statements are equivalent; (i) X f ⊆ ∪ α∈S X g α onlyif X f ⊆ X g α for some α ∈ S, (ii) V(f) ⊆ ∪ α∈S V (g α ) only ifV (f) ⊆ V (g α ) for some α ∈ S, (iii) V (f) ⊆ ∪ ni=1 V (g i ) only ifV (f) ⊆ V (g i ) for some i, (iv) Spec(R) is linearly ordered underinclusion. Let R be a commutative ring (with identity 1). It is known thatevery prime ideal of R is the radical of a principal ideal if and onlyif R satisfies the following property; (∗) for a prime ideal P and a(nonempty) set {P α |α ∈ A} of prme ideals of R, P ⊆ ∪ α∈A P α impliesP ⊆ P α for some α ∈ A. This was proved for Noetherian rings byReis and Viswanathan [6], and then completely generalized by Smith[7]. It is clear that if R satisfies (∗), then R has a finite numberof minimal prime ideals (cf. [2, Theorem 2.1] or [4, Theorem 2.5]).As a natural dual of (∗), Gilmer [3] studied the following condition;(#) If P ∈ Spec(R) and if {I" @default.
- W318703076 created "2016-06-24" @default.
- W318703076 creator A5075038663 @default.
- W318703076 date "2005-01-01" @default.
- W318703076 modified "2023-09-28" @default.
- W318703076 title "A COVERING CONDITION FOR THE PRIME SPECTRUMS" @default.
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