Δ-closures of ideals and rings

Abstract

Abstract.It is shown that if R is a commutative ring with identity and A is a multiplicatively closed set of finitely generated nonzero ideals of R , then the operation / -» IA = [JKeA(IK '■ K) is a closure operation on the set of ideals / of R that satisfies a partial cancellation law, and it is a prime operation if and only if R is A-closed. Also, if none of the ideals in A is contained in a minimal prime ideal, then IA Ç Ia , the integral closure of / in R , and if A is the set of all such finitely generated ideals and / contains an ideal in A , then IA = /" . Further, R has a natural A-closure RA , A -> AA is a closure operation on a large set of rings A that contain R as a subring, A -► AA behaves nicely under certain types of ring extension, and every integral extension overring of R is RA for an appropriate set A . Finally, if R is Noetherian, then the associated primes of IA are also associated primes of IAK and (IK)A for all K e A.

A Guide to Closure Operations in Commutative Algebra

“…[Bir67, V.1] or [DP02, 7.1]). Oddly, this general definition of closure operation does not seem to have gained currency in commutative algebra until the late 1980s [Rat89,OR88], although more special structures already had standard terminologies associated to them (see 4.1).…”

“…We define solid closure on R by saying that x ∈ I ⋆ if x ∈ IS for some solid R-algebra S. (See 3.2(9) for general R.) (10) Let ∆ be a multiplicatively closed set of ideals. The ∆-closure [Rat89] of an ideal I is I ∆ := K∈∆ (IK : K). Ratliff [Rat89] shows close connections between ∆-closure and integral closure for appropriate choices of ∆.…”

Δ-Reductions and Δ-Closures of Ideals with Respect to an Artinian Module

“…These concepts have been extended to ideals in an arbitrary commutative ring by L. J. Ratliff. Namely, the important notions of -closure and -reduction of an ideal in a commutative ring R were introduced and studied in Ratliff (1989) and Ratliff and Rush (1993) as a refinement of the reduction and integral closure of an ideal, and these new ideas have been proven useful in several questions, for example see 764 NAGHIPOUR AND SEDGHI Huneke (1987), McAdam (1983), and Ratliff and Rush (2002). It is appropriate for us to provide a brief review.…”

Contracted, -Full and Related Classes of Ideals in Local Rings

“…For some further examples and applications of closure operations, see for example [3,16,17]. We use the following terminology from [10,11,13].…”

“…EXAMPLE 2.3. The -closure is a semi-prime operation [11]. Let R be a commutative ring with identity and a multiplicatively closed set of non-zero finitely generated ideals of R. If I is an ideal in R, then D(I) = {IK : R K | K ∈ } is a directed set and ∪{IK : R K | K ∈ } = K∈ (IK : R K) is an ideal I called the delta-closure of I.…”