MULTIPLICATIVELY CLOSED SETS OF IDEALS AND RESIDUAL DIVISION

Abstract

Let R be a ring (commutative with identity), let Γ be a multiplicatively closed set of finitely generated nonzero ideals of R, for an ideal I of R let I Γ = ∪ {I : G; G ∈ Γ}, and for an R-algebra A such that GA ≠ (0) for all G ∈ Γ let A Γ = ∪ {A : T GA; G ∈ Γ}, where T is the total quotient ring of A. Then I Γ is an ideal in R, I → I Γ is a semi-prime operation (on the set I of ideals I of R) that satisfies a cancellation law, and it is a prime operation on I if and only if R = R Γ. Also, A Γ is an R-algebra and A → A Γ is a closure operation on any set A = {A; A is an R-algebra, R ⊆ A, and if C is a ring between R and A, then regular elements in C remain regular in A}. Finally, several results are proved concerning relations between the ideals I Γ and (IA)ΓA and between the R-algebras R Γ and A Γ. *Ahmad died of a heart attack on January 28, 2000. He was the gentlest, kindest, most humble of men. -LJR