WEAKLY PRIMARY IDEAL AND Φ-PRIMARY IDEAL IN AMALGAMATED ALGEBRA ALONG AN IDEAL

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text], [Formula: see text] an ideal of [Formula: see text], [Formula: see text] an [Formula: see text]-primary ideal of [Formula: see text], [Formula: see text] a finitely generated [Formula: see text]-module, [Formula: see text] a finitely generated standard graded algebra over [Formula: see text] and [Formula: see text] a finitely generated graded [Formula: see text]-module. We characterize the multiplicity and the Cohen–Macaulayness of the fiber cone [Formula: see text]. As an application, we obtain some results on the multiplicity and the Cohen–Macaulayness of the fiber cone[Formula: see text].

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Reduction numbers and Hilbert polynomials of ideals in higher dimensional CohenMacaulay local rings

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100075149 ◽

1992 ◽

Vol 111 (1) ◽

pp. 47-56 ◽

Cited By ~ 4

Author(s):

Yinghwa Wu

Keyword(s):

Local Ring ◽

Residue Field ◽

Primary Ideal ◽

Local Rings ◽

Hilbert Polynomials ◽

Higher Dimensional ◽

Reduction Number ◽

Minimal Reduction ◽

System Of Parameters

Throughout, (R, m) will denote a d-dimensional CohenMacaulay (CM for short) local ring having an infinite residue field and I an m-primary ideal in R. Recall that an ideal J I is said to be a reduction of I if Ir+1 = JIr for some r 0, and a reduction J of I is called a minimal reduction of I if J is generated by a system of parameters. The concepts of reduction and minimal reduction were first introduced by Northcott and Rees12. If J is a reduction of I, define the reduction number of I with respect to J, denoted by rj(I), to be min {r 0 Ir+1 = JIr}. The reduction number of I is defined as r(I) = min {rj(I)J is a minimal reduction of I}. The reduction number r(I) is said to be independent if r(I) = rj(I) for every minimal reduction J of I.

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Artinianness of Composed Graded Local Cohomology Modules

Canadian Mathematical Bulletin ◽

10.4153/cmb-2016-006-6 ◽

2016 ◽

Vol 59 (2) ◽

pp. 271-278

Author(s):

Fatemeh Dehghani-Zadeh

Keyword(s):

Noetherian Ring ◽

Local Cohomology ◽

Primary Ideal ◽

Finitely Generated ◽

Local Cohomology Modules ◽

Base Ring ◽

Cohomology Modules ◽

Local Base

AbstractLet be a graded Noetherian ring with local base ring (R0 ,m0) and let . Let M and N be finitely generated graded R-modules and let a = a0 + R+ an ideal of R. We show that and are Artinian for some i s and j s with a specified property, where bo is an ideal of R0 such that a0 + b0 is an m0-primary ideal.

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Positive deissler rank and the complexity of injective modules

Journal of Symbolic Logic ◽

10.1017/s0022481200029108 ◽

1988 ◽

Vol 53 (1) ◽

pp. 284-293 ◽

Cited By ~ 1

Author(s):

T. G. Kucera

Keyword(s):

Prime Ideal ◽

Lower Bounds ◽

Upper Bound ◽

Noetherian Ring ◽

Krull Dimension ◽

Primary Ideal ◽

Prime Ideals ◽

Commutative Noetherian Ring ◽

Injective Modules ◽

Primary Ideals

This is the second of two papers based on Chapter V of the author's Ph.D. thesis [K1]. For acknowledgements please refer to [K3]. In this paper I apply some of the ideas and techniques introduced in [K3] to the study of a very specific example. I obtain an upper bound for the positive Deissler rank of an injective module over a commutative Noetherian ring in terms of Krull dimension. The problem of finding lower bounds is vastly more difficult and is not addressed here, although I make a few comments and a conjecture at the end.For notation, terminology and definitions, I refer the reader to [K3]. I also use material on ideals and injective modules from [N] and [Ma]. Deissler's rank was introduced in [D].For the most part, in this paper all modules are unitary left modules over a commutative Noetherian ring Λ but in §1 I begin with a few useful results on totally transcendental modules and the relation between Deissler's rank rk and rk+.Recall that if P is a prime ideal of Λ, then an ideal I of Λ is P-primary if I ⊂ P, λ ∈ P implies that λn ∈ I for some n and if λµ ∈ I, λ ∉ P, then µ ∈ I. The intersection of finitely many P-primary ideals is again P-primary, and any P-primary ideal can be written as the intersection of finitely many irreducible P-primary ideals since Λ is Noetherian. Every irreducible ideal is P-primary for some prime ideal P. In addition, it will be important to recall that if P and Q are prime ideals, I is P-primary, J is Q-primary, and J ⊃ I, then Q ⊃ P. (All of these results can be found in [N].)

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On Nagata rings with amalgamation properties

Asian-European Journal of Mathematics ◽

10.1142/s1793557121501369 ◽

2020 ◽

pp. 2150136

Author(s):

Ali Molkhasi ◽

Kar Ping Shum

Keyword(s):

Commutative Rings ◽

Ring Homomorphism ◽

Nagata Ring ◽

Amalgamated Algebra

Let [Formula: see text] and [Formula: see text] be two commutative rings with unity, let [Formula: see text] be an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. In this paper, we give a characterization for the amalgamated algebra [Formula: see text] to be a Nagata ring, a strong S-domain, and a catenarian. Also, we investigate the conditions that the ring of Hurwitz series over [Formula: see text] has a complete comaximal factorization.

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