Associated primes and integral closure of Noetherian rings

Let E be an injective module over a commutative Noetherian ring A (with non-zero identity), and let a be an ideal of A. The submodule (0:Eα) of E has a secondary representation, and so we can form the finite set AttA(0:Eα) of its attached prime ideals. In [1, 3.1], we showed that the sequence of sets is ultimately constant; in [2], we introduced the integral closure a*(E) of α relative to E, and showed that is increasing and ultimately constant. In this paper, we prove that, if a contains an element r such that rE = E, then is ultimately constant, and we obtain information about its ultimate constant value.

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The Quotient Problem for Noetherian Rings

Canadian Journal of Mathematics ◽

10.4153/cjm-1981-056-5 ◽

1981 ◽

Vol 33 (3) ◽

pp. 734-748 ◽

Cited By ~ 7

Author(s):

Bruno J. Müller

Keyword(s):

Noetherian Ring ◽

Quotient Ring ◽

Noetherian Rings ◽

Associated Primes ◽

Singular Proposition ◽

Role Taking ◽

Composition Factors

Our work was motivated by attempts to find a criterion for the existence of a classical quotient ring, for a noetherian ring, in analogy with the various known criteria for the existence of an artinian classical quotient ring ([9], [10], [13], [2]).We have restricted our attention to Krull symmetric noetherian rings R, and we make heavy use of the fact that all their Krull composition factors are non-singular (Proposition 7). The collection Kprime R of the associated primes of the Krull composition factors of R plays a central role, taking the place of the collection of the associated primes of R.

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Integral closure of Noetherian rings

Proceedings of the 1997 international symposium on Symbolic and algebraic computation - ISSAC '97 ◽

10.1145/258726.258786 ◽

1997 ◽

Cited By ~ 4

Author(s):

Patrizia Gianni ◽

Barry Trager

Keyword(s):

Integral Closure ◽

Noetherian Rings

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U–essential prime divisors and sequences over an ideal

Nagoya Mathematical Journal ◽

10.1017/s002776300000057x ◽

1986 ◽

Vol 103 ◽

pp. 39-66 ◽

Cited By ~ 19

Author(s):

Daniel Katz ◽

Louis J. Ratliff

Keyword(s):

Prime Ideal ◽

Noetherian Ring ◽

Asymptotic Theory ◽

Minimal Prime Ideal ◽

Standard Theory ◽

Noetherian Rings ◽

Associated Primes ◽

Prime Divisors ◽

Minimal Prime

All rings in this paper are assumed to be commutative with identity, and they will generally also be Noetherian.In several recent papers the asymptotic theory of ideals in Noetherian rings has been introduced and developed. In this new theory the roles played in the standard theory by associated primes, R-sequences, classical grade, and Cohen-Macaulay rings are played by, respectively, asymptotic prime divisors, asymptotic sequences, asymptotic grade, and locally quasi-unmixed Noetherian rings. And up to the present time it has been shown that quite a few results from the standard theory have a valid analogue in the asymptotic theory, and a number of interesting and useful new results concerning the asymptotic prime divisors of an ideal in a Noetherian ring have also been proved. In fact the analogy between the two theories is so good that a very useful (but not completely valid) working guide is: results from the standard theory should have a valid analogue in the asymptotic theory. And, although asymptotic sequences are coarser than R-sequences (for example, they behave nicely when passing to R/z with z a minimal prime ideal in R), the converse of this working guide has also proved useful.

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Limit behavior of the rational powers of monomial ideals

Journal of Algebra and Its Applications ◽

10.1142/s021949882350069x ◽

2021 ◽

Author(s):

James Lewis

Keyword(s):

Local Cohomology ◽

Integral Closure ◽

Monomial Ideals ◽

Limit Behavior ◽

Associated Primes ◽

Powers Of Ideals ◽

Symbolic Powers ◽

Local Cohomology Modules ◽

Cohomology Modules

We investigate the rational powers of ideals. We find that in the case of monomial ideals, the canonical indexing leads to a characterization of the rational powers yielding that symbolic powers of squarefree monomial ideals are indeed rational powers themselves. Using the connection with symbolic powers techniques, we use splittings to show the convergence of depths and normalized Castelnuovo–Mumford regularities. We show the convergence of Stanley depths for rational powers, and as a consequence of this, we show the before-now unknown convergence of Stanley depths of integral closure powers. Additionally, we show the finiteness of asymptotic associated primes, and we find that the normalized lengths of local cohomology modules converge for rational powers, and hence for symbolic powers of squarefree monomial ideals.

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Associated Prime Ideals in Non-Noetherian Rings

Canadian Journal of Mathematics ◽

10.4153/cjm-1984-021-6 ◽

1984 ◽

Vol 36 (2) ◽

pp. 344-360 ◽

Cited By ~ 10

Author(s):

Juana Iroz ◽

David E. Rush

Keyword(s):

Prime Ideal ◽

Finite Subset ◽

Noetherian Ring ◽

Prime Ideals ◽

Noetherian Rings ◽

Associated Primes ◽

The Past ◽

Associated Prime Ideals ◽

Associated Prime

The theory of associated prime ideals is one of the most basic notions in the study of modules over commutative Noetherian rings. For modules over non-Noetherian rings however, the classical associated primes are not so useful and in fact do not exist for some modules M. In [4] [22] a prime ideal P of a ring R is said to be attached to an R-module M if for each finite subset I of P there exists m ∊ M such that I ⊆ annR(m) ⊆ P. In [4] the attached primes were compared to the associated primes and the results of [4], [22], [23], [24] show that the attached primes are a useful alternative in non-Noetherian rings to associated primes. Several other methods of associating a set of prime ideals to a module M over a non-Noetherian ring have proven very useful in the past. The most common of these is the set Assf(M) of weak Bourbaki primes of M [2, pp. 289-290].

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On prime divisors of large powers of elements in Noether lattices

Glasgow Mathematical Journal ◽

10.1017/s0017089500007461 ◽

1988 ◽

Vol 30 (3) ◽

pp. 359-367 ◽

Cited By ~ 1

Author(s):

Linda Becerra

Keyword(s):

Noetherian Ring ◽

Integral Closure ◽

Noetherian Rings ◽

Prime Divisors ◽

Commutative Noetherian Ring ◽

Fixed Element ◽

Fixed Set ◽

Decomposition Theorems ◽

Lattice Of Ideals ◽

Multiplicative Lattices

In [4], R. P. Dilworth introduced the concept of a Noether lattice as an abstraction of the lattice of ideals of a Noetherian ring and he showed that many important properties of Noetherian rings, such as the Noether decomposition theorems, also hold for Noether lattices. It was later shown, in [1], that every Noether lattice is not the lattice of ideals of any Noetherian ring, yet many studies have successfully been undertaken to relate other concepts between Noetherian rings and Noether lattices as had been begun by Dilworth. (See [3], [5], and [6].) In this paper we undertake such a study and show that some results of M. Brodmann in [2] and L. Ratliff in [7] concerning prime divisors of large powers of a fixed element of a commutative Noetherian ring may be generalized and extended to the setting of a Noether lattice. It is shown (Theorem 2.8) that if A is an element of a Noether lattice then all large powers of A have the same prime divisors and (Corollary 3.8) included among this fixed set of primes are those primes that are prime divisors of the integral closure of Ak for some k≧l. We note that the ring proof of this latter result does not generalize directly since it uses the notion of transcendence degree which to our knowledge has no analogue in multiplicative lattices.

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