REMARKS ON THE THEORY OF ATTACHED PRIME IDEALS

1982 ◽  
Vol 33 (2) ◽  
pp. 239-245
Author(s):  
D. G. NORTHCOTT
Keyword(s):  
Prime Ideals ◽  
Attached Prime

scholarly journals On the attached prime ideals of localcohomology modules defined by a pair of ideals

2017 ◽  
Vol 41 ◽  
pp. 216-222
Author(s):  
Zohreh HABIBI ◽  
Maryam JAHANGIRI ◽  
Khadijeh AHMADI AMOLI
Keyword(s):  
Prime Ideals ◽  
Attached Prime

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.

Asymptotic attached prime ideals related to injective modules

1992 ◽  
Vol 20 (2) ◽  
pp. 583-590 ◽  
Author(s):  
Leif Melkersson ◽  
Peter Schenzel
Keyword(s):  
Prime Ideals ◽  
Injective Modules ◽  
Attached Prime

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 The finiteness of certain sets of attached prime ideals and the length of generalized fractions

2004 ◽  
Vol 189 (1-3) ◽  
pp. 109-121 ◽  
Author(s):  
Nguyen Tu Cuong ◽  
Marcel Morales ◽  
Le Thanh Nhan
Keyword(s):  
Prime Ideals ◽  
Attached Prime

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.

ATTACHED PRIMES UNDER SKEW POLYNOMIAL EXTENSIONS

2011 ◽  
Vol 10 (03) ◽  
pp. 537-547 ◽  
Author(s):  
SCOTT ANNIN
Keyword(s):  
Polynomial Ring ◽  
Natural Generalization ◽  
Polynomial Rings ◽  
Prime Ideals ◽  
Associated Primes ◽  
Noncommutative Rings ◽  
Artinian Modules ◽  
Skew Polynomial Rings ◽  
Skew Polynomial ◽  
Attached Prime

In the author's work [S. A. Annin, Attached primes over noncommutative rings, J. Pure Appl. Algebra212 (2008) 510–521], a theory of attached prime ideals in noncommutative rings was developed as a natural generalization of the classical notions of attached primes and secondary representations that were first introduced in 1973 as a dual theory to the associated primes and primary decomposition in commutative algebra (see [I. G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math.11 (1973) 23–43]). Associated primes over noncommutative rings have been thoroughly studied and developed for a variety of applications, including skew polynomial rings: see [S. A. Annin, Associated primes over skew polynomial rings, Commun. Algebra30(5) (2002) 2511–2528; and S. A. Annin, Associated primes over Ore extension rings, J. Algebra Appl.3(2) (2004) 193–205]. Motivated by this background, the present article addresses the behavior of the attached prime ideals of inverse polynomial modules over skew polynomial rings. The goal is to determine the attached primes of an inverse polynomial module M[x-1] over a skew polynomial ring R[x;σ] in terms of the attached primes of the base module MR. This study was completed in the commutative setting for the class of representable modules in [L. Melkersson, Content and inverse polynomials on artinian modules, Commun. Algebra26(4) (1998) 1141–1145], and the generalization to noncommutative rings turns out to be quite non-trivial in that one must either work with a Bass module MR or a right perfect ring R in order to achieve the desired statement even when no twist is present in the polynomial ring "Let MR be a module over any ring R. If M[x-1]R is a completely σ-compatible Bass module, then Att (M[x-1]S) = {𝔭[x] : 𝔭 ∈ Att (MR)}." The sharpness of the results are illustrated through the use of several illuminating examples.

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