scholarly journals On prime divisors of large powers of elements in Noether lattices

1988 ◽  
Vol 30 (3) ◽  
pp. 359-367 ◽  
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.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings II

1992 ◽  
Vol 35 (3) ◽  
pp. 511-518
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Integral Closure ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

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.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings

1991 ◽  
Vol 34 (1) ◽  
pp. 155-160 ◽  
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

LetEbe an injective module over the commutative Noetherian ringA, and letabe an ideal ofA. TheA-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈Nis ultimately constant. This result is analogous to a theorem of M. Brodmann that, ifMis a finitely generatedA-module, then the sequence of sets (AssA(M/αnM))n∈Nis ultimately constant.

Bounded Derived Categories of Infinite Quivers: Grothendieck Duality, Reflection Functor

2015 ◽  
Vol 67 (1) ◽  
pp. 28-54 ◽  
Author(s):  
Javad Asadollahi ◽  
Rasool Hafezi ◽  
Razieh Vahed
Keyword(s):  
Noetherian Ring ◽  
Global Dimension ◽  
Derived Categories ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
Finite Global Dimension

AbstractWe study bounded derived categories of the category of representations of infinite quivers over a ring R. In case R is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left (resp. right) rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.

scholarly journals On commutative Noetherian rings which satisfy the radical formula

1997 ◽  
Vol 39 (3) ◽  
pp. 285-293 ◽  
Author(s):  
Ka Hin Leung ◽  
Shing Hing Man
Keyword(s):  
Noetherian Ring ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring

AbstractIn this paper, we show that a commutative Noetherian ring which satisfies the radical formula must be of dimension at most one. From this we give a characterization of commutative Noetherian rings that satisfy the radical formula.

Form rings and projective equivalence

1986 ◽  
Vol 99 (3) ◽  
pp. 447-456 ◽  
Author(s):  
Daniel Katz ◽  
L. J. Ratliff
Keyword(s):  
Local Ring ◽  
Prime Divisor ◽  
Noetherian Ring ◽  
Noetherian Rings ◽  
Local Rings ◽  
Prime Divisors ◽  
Research Problems ◽  
Form Ring ◽  
Divisor Of Zero ◽  
Minimal Prime

If I and J are ideals in a Noetherian ring R, then I and J are projectively equivalent in case (Ii)a = (Jj)a for some positive integers i, j (where Ka denotes the integral closure in R of the ideal K) and the form ring F(R, I) of R with respect to I is the graded ring R/I ⊕ I/I2 ⊕ I2/I3 ⊕ …. These two concepts have played an important role in many research problems in commutative algebra, so they have been deeply studied and many of their properties have been discovered. In a recent paper [13] they were combined to show that a semi-local ring R is unmixed if and only if for every ideal J in R there exists a projectively equivalent ideal J in R such that every prime divisor of zero in F(R, J) has the same depth. It seems to us that results similar to this are interesting and potentially quite useful, so in this paper we prove several additional such theorems. Namely, it is shown that all ideals in all local rings have a projectively equivalent ideal whose form ring is fairly nice. Also, a characterization similar to the just mentioned result in [13] is given for the class of local rings whose completions have no embedded prime divisors of zero, and several analogous new characterizations are given for locally unmixed Noetherian rings. In particular, it is shown that if I is an ideal in an unmixed local ring R such that height(I) = l(I) (where l(I) denotes the analytic spread of I), then there exists a projectively equivalent ideal J in R such that Ass (F(R, J)) has exactly m elements, all minimal, where m is the number of minimal prime divisors of I (so if I is open, then F(R, J) has exactly one prime divisor of zero and is a locally unmixed Noetherian ring).

scholarly journals On asymptotic values of certain sets of attached prime ideals

1988 ◽  
Vol 30 (3) ◽  
pp. 293-300 ◽  
Author(s):  
A.-J. Taherizadeh
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Primary Decomposition ◽  
Prime Divisors ◽  
Finitely Generated Module ◽  
Commutative Noetherian Ring ◽  
Asymptotic Values ◽  
Attached Prime ◽  
Positive Integers

In his paper [1], M. Brodmann showed that if M is a1 finitely generated module over the commutative Noetherian ring R (with identity) and a is an ideal of R then the sequence of sets {Ass(M/anM)}n∈ℕ and {Ass(an−1M/anM)}n∈ℕ (where ℕ denotes the set of positive integers) are eventually constant. Since then, the theory of asymptotic prime divisors has been studied extensively: in [5], Chapters 1 and 2], for example, various results concerning the eventual stable values of Ass(R/an;) and Ass(an−1/an), denoted by A*(a) and B*(a) respectively, are discussed. It is worth mentioning that the above mentioned results of Brodmann still hold if one assumes only that A is a commutative ring (with identity) and M is a Noetherian A-module, and AssA(M), in this situation, is regarded as the set of prime ideals belonging to the zero submodule of M for primary decomposition.

Complexes acyclic up to integral closure

1994 ◽  
Vol 116 (3) ◽  
pp. 401-414
Author(s):  
D. Katz
Keyword(s):  
Local Ring ◽  
Homomorphic Image ◽  
Noetherian Ring ◽  
Integral Closure ◽  
Koszul Complex ◽  
Gorenstein Ring ◽  
Tight Closure ◽  
Commutative Noetherian Ring ◽  
Remarkable Result ◽  
Depth Sensitivity

In [R] D. Rees introduced the notions of reduction and integral closure for modules over a commutative Noetherian ring and proved the following remarkable result. Let R be a locally quasi-unmixed Noetherian ring and I an ideal generated by n elements. Suppose that height (I) = h. Then the ith module of cycles in the Koszul complex on a set of n generators for I is contained in the integral closure of the ith module of boundaries for i > n − h. This result should be considered a dimension-theoretic analogue of the famous depth sensitivity property of the Koszul complex demonstrated by Serre and Auslander-Buschsbaum in the 1950s. At roughly the same time, Hoschster and Huneke introduced the notion of tight closure and thereafter gave a number of theorems in the same (though considerably broader) vein for tight closure. In particular, in [HH] they showed that if R is an equidimensional local ring of characteristic p > 0, which is a homomorphic image of a Gorenstein ring, then for all i > 0, the ith module of cycles is contained in the tight closure of the ith module of boundaries for any complex satisfying the so-called standard rank and height conditions (see the definitions below). Since the tight closure is contained in the integral closure for such rings, the result of Hochster and Huneke extends (in characteristic p) considerably the result of Rees. In fact, their result could be considered a dimension-theoretic analogue of the Buchsbaum-Eisenbud exactness theorem ([BE]), which in a certain sense is the ultimate depth sensitivity theorem. Moreover, using the technique of reduction to characteristic p, Hochster and Huneke have shown that their results hold in equicharacteristic zero as well, whenever the tight closure is defined.

scholarly journals Totally acyclic complexes and locally Gorenstein rings

2018 ◽  
Vol 17 (03) ◽  
pp. 1850039
Author(s):  
Lars Winther Christensen ◽  
Kiriko Kato
Keyword(s):  
Noetherian Ring ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
Gorenstein Rings ◽  
Dualizing Complex ◽  
Injective Modules ◽  
Acyclic Complex ◽  
Acyclic Complexes

A commutative noetherian ring with a dualizing complex is Gorenstein if and only if every acyclic complex of injective modules is totally acyclic. We extend this characterization, which is due to Iyengar and Krause, to arbitrary commutative noetherian rings, i.e. we remove the assumption about a dualizing complex. In this context Gorenstein, of course, means locally Gorenstein at every prime.

Krull Dimension of Injective Modules Over Commutative Noetherian Rings

2005 ◽  
Vol 48 (2) ◽  
pp. 275-282
Author(s):  
Patrick F. Smith
Keyword(s):  
Noetherian Ring ◽  
Integral Domain ◽  
Krull Dimension ◽  
Injective Module ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
One Dimensional ◽  
Injective Modules

AbstractLet R be a commutative Noetherian integral domain with field of fractions Q. Generalizing a forty-year-old theorem of E. Matlis, we prove that the R-module Q/R (or Q) has Krull dimension if and only if R is semilocal and one-dimensional. Moreover, if X is an injective module over a commutative Noetherian ring such that X has Krull dimension, then the Krull dimension of X is at most 1.

scholarly journals U–essential prime divisors and sequences over an ideal

1986 ◽  
Vol 103 ◽  
pp. 39-66 ◽  
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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