Some results on the asymptotic completion of an ideal

“…However, in both [4] and [10] it is pointed out that all prime operations that satisfy these additional properties are essentially known. Since we want to derive a new family of prime operations in this paper, we chose the definition in (2.3), which does agree with that given in [6] and [8].…”

Section: The A-closure Of An Idealmentioning

confidence: 99%

“…Closure operations (see (2.3) for the definitions) are of some interest in themselves, and they have been studied both in relation to specific ideal-closures (such as integral closure) and abstractly; for example, see [2,3,4,§43,6,8,10,Appendix 4]. In this paper a new family of closure operations is introduced and several of the classical properties of integral closure are extended to these new closures.…”

Section: Introductionmentioning

confidence: 99%

Δ-closures of ideals and rings

Ratliff¹

1989

Trans. Amer. Math. Soc.

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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.

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“…The above theorem gives one such property, and that such rings have this property is a new result, and until now was not known to hold even in a regular local ring or a complete local domain. Since the concept of the integral closure of an ideal has proved to beu seful in numerous research papers, for example [1], [2], [5], [6], [14], [15], [16], [17], and [20], the fact that ideals of the principal class in such rings have the property stated in the theorem should be of importance in future investigations. On the other hand, there are many open problems concerning the chain conjectures (see [11, § 2] for the relationship of quasi-unmixedness to the chain conditions for prime ideals, and see the introduction of [12] for some of the open problems in this area), so properties of locally quasi-unmixed Noetherian rings which characterize such rings are of interest and importance in solving these problems.…”

mentioning

confidence: 99%

“…^ (((#0.)0. = (^%0α (since I-+I a is a semiprime operation on the set of ideals in R [6]). Therefore, the conclusion follows from 2.12.…”

mentioning

confidence: 99%