Quasi-primaryful modules

2015 ◽  
Vol 08 (03) ◽  
pp. 1550051
Author(s):  
Hosein Fazaeli Moghimi ◽  
Mahdi Samiei
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Artinian Ring ◽  
The Other ◽  
Noetherian Rings ◽  
Projective Modules ◽  
New Class ◽  
The Family ◽  
Free Modules ◽  
Finitely Generated Modules

Let [Formula: see text] be a commutative ring with identity. The purpose of this paper is to introduce and to study a new class of modules over [Formula: see text] called quasi-primaryful [Formula: see text]-modules. This class contains the family of finitely generated modules properly, on the other hand it is contained in the family of primeful [Formula: see text]-modules properly, and three concepts coincide if they are multiplication modules. We show that free modules, projective modules over domains and faithful projective modules over Noetherian rings are quasi-primaryful modules. In particular, if [Formula: see text] is an Artinian ring, then all [Formula: see text]-modules are quasi-primaryful and the converse is also true when [Formula: see text] is a Noetherian ring.

Strict Mittag–Leffler modules over Gorenstein injective modules

2019 ◽  
Vol 19 (03) ◽  
pp. 2050050 ◽  
Author(s):  
Yanjiong Yang ◽  
Xiaoguang Yan
Keyword(s):  
Noetherian Ring ◽  
Direct Limit ◽  
Injective Module ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Injective Modules ◽  
Pure Submodules ◽  
Covering Class ◽  
Gorenstein Injective ◽  
Finitely Presented

In this paper, we study the conditions under which a module is a strict Mittag–Leffler module over the class [Formula: see text] of Gorenstein injective modules. To this aim, we introduce the notion of [Formula: see text]-projective modules and prove that over noetherian rings, if a module can be expressed as the direct limit of finitely presented [Formula: see text]-projective modules, then it is a strict Mittag–Leffler module over [Formula: see text]. As applications, we prove that if [Formula: see text] is a two-sided noetherian ring, then [Formula: see text] is a covering class closed under pure submodules if and only if every injective module is strict Mittag–Leffler over [Formula: see text].

scholarly journals The Ohm–Rush content function II. Noetherian rings, valuation domains, and base change

2019 ◽  
Vol 18 (05) ◽  
pp. 1950100
Author(s):  
Neil Epstein ◽  
Jay Shapiro
Keyword(s):  
Noetherian Ring ◽  
Artinian Ring ◽  
Base Change ◽  
Noetherian Rings ◽  
Prime Characteristic ◽  
Field Extensions ◽  
Dimension Formula ◽  
Valuation Domains ◽  
Prüfer Domains ◽  
Polynomial Extensions

The notion of an Ohm–Rush algebra, and its associated content map, has connections with prime characteristic algebra, polynomial extensions, and the Ananyan–Hochster proof of Stillman’s conjecture. As further restrictions are placed (creating the increasingly more specialized notions of weak content, semicontent, content, and Gaussian algebras), the construction becomes more powerful. Here we settle the question in the affirmative over a Noetherian ring from [N. Epstein and J. Shapiro, The Ohm-Rush content function, J. Algebra Appl. 15(1) (2016) 1650009, 14 pp.] of whether a faithfully flat weak content algebra is semicontent (and over an Artinian ring of whether such an algebra is content), though both questions remain open in general. We show that in content algebra maps over Prüfer domains, heights are preserved and a dimension formula is satisfied. We show that an inclusion of nontrivial valuation domains is a content algebra if and only if the induced map on value groups is an isomorphism, and that such a map induces a homeomorphism on prime spectra. Examples are given throughout, including results that show the subtle role played by properties of transcendental field extensions.

scholarly journals Commutative rings whose quotients are Goldie

1975 ◽  
Vol 16 (1) ◽  
pp. 32-33 ◽  
Author(s):  
Victor P. Camillo
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Quotient Ring ◽  
Chain Condition ◽  
Commutative Rings ◽  
The Other ◽  
Direct Sums ◽  
Other Hand ◽  
Commutative Domain ◽  
Ascending Chain Condition

All rings considered here have units. A (non-commutative) ring is right Goldieif it has no infinite direct sums of right ideals and has the ascending chain condition on annihilator right ideals. A right ideal A is an annihilator if it is of the form {a ∈ R/xa = 0 for all x ∈ X}, where X is some subset of R. Naturally, any noetherian ring is Goldie, but so is any commutative domain, so that the converse is not true. On the other hand, since any quotient ring of a noetherian ring is noetherian, it is true that every quotient is Goldie. A reasonable question therefore is the following: must a ring, such that every quotient ring is Goldie, be noetherian? We prove the following theorem:Theorem. A commutative ring is noetherian if and only if every quotient is Goldie.

scholarly journals Pseudo-Noetherian Rings

1976 ◽  
Vol 19 (1) ◽  
pp. 77-84 ◽  
Author(s):  
Kenneth P. McDowell
Keyword(s):  
Commutative Ring ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Finitely Generated Modules

In the latter part of the 1950’s some interesting papers appeared (e.g. [2] and [10]) which examined the relationships occurring between the purely algebraic and homological aspects of the theory of finitely generated modules over Noetherian rings. Many of these relationships remain valid if one considers the much wider class of rings determined by the following definition.Definition. A commutative ring R is called pseudo-Noetherian if it satisfies the following two conditions.

scholarly journals On weakly S-prime ideals of commutative rings

2021 ◽  
Vol 29 (2) ◽  
pp. 173-186
Author(s):  
Fuad Ali Ahmed Almahdi ◽  
El Mehdi Bouba ◽  
Mohammed Tamekkante
Keyword(s):  
Prime Ideal ◽  
Commutative Ring ◽  
Principal Ideal ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
New Class ◽  
Weakly Prime Ideals

Abstract Let R be a commutative ring with identity and S be a multiplicative subset of R. In this paper, we introduce the concept of weakly S-prime ideals which is a generalization of weakly prime ideals. Let P be an ideal of R disjoint with S. We say that P is a weakly S-prime ideal of R if there exists an s ∈ S such that, for all a, b ∈ R, if 0 ≠ ab ∈ P, then sa ∈ P or sb ∈ P. We show that weakly S-prime ideals have many analog properties to these of weakly prime ideals. We also use this new class of ideals to characterize S-Noetherian rings and S-principal ideal rings.

scholarly journals A Note on Weakly S-Noetherian Rings

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 419 ◽  
Author(s):  
Dong Kyu Kim ◽  
Jung Wook Lim
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Proper Ideal ◽  
Noetherian Rings ◽  
Amalgamated Algebra

Let R be a commutative ring with identity and S a (not necessarily saturated) multiplicative subset of R. We call the ring R to be a weakly S-Noetherian ring if every S-finite proper ideal of R is an S-Noetherian R-module. In this article, we study some properties of weakly S-Noetherian rings. In particular, we give some conditions for the Nagata’s idealization and the amalgamated algebra to be weakly S-Noetherian rings.

scholarly journals Dimension of finite free complexes over commutative Noetherian rings

2021 ◽  
pp. 11-17
Author(s):  
Lars Christensen ◽  
Srikanth Iyengar
Keyword(s):  
Finite Rank ◽  
Noetherian Ring ◽  
Krull Dimension ◽  
Noetherian Rings ◽  
Commutative Noetherian Ring ◽  
Bounded Complex ◽  
Free Modules

Foxby defined the (Krull) dimension of a complex of modules over a commutative Noetherian ring in terms of the dimension of its homology modules. In this note it is proved that the dimension of a bounded complex of free modules of finite rank can be computed directly from the matrices representing the differentials of the complex.

scholarly journals Monomial multiplicities in explicit form

2019 ◽  
Vol 19 (09) ◽  
pp. 2050181
Author(s):  
Guillermo Alesandroni
Keyword(s):  
Explicit Formula ◽  
Explicit Form ◽  
Complete Intersection ◽  
Polynomial Ring ◽  
Monomial Ideal ◽  
The Other ◽  
Complete Intersections ◽  
New Class ◽  
Other Hand ◽  
The Family

Denote by [Formula: see text] a polynomial ring over a field, and let [Formula: see text] be a monomial ideal of [Formula: see text]. If [Formula: see text], we prove that the multiplicity of [Formula: see text] is given by [Formula: see text] On the other hand, if [Formula: see text] is a complete intersection, and [Formula: see text] is an almost complete intersection, we show that [Formula: see text] We also introduce a new class of ideals that extends the family of monomial complete intersections and that of codimension 1 ideals, and give an explicit formula for their multiplicity.

scholarly journals On Nonnil-S-Noetherian Rings

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1428 ◽  
Author(s):  
Min Jae Kwon ◽  
Jung Wook Lim
Keyword(s):  
Power Series ◽  
Commutative Ring ◽  
Polynomial Ring ◽  
Noetherian Ring ◽  
Type Theorem ◽  
Noetherian Rings ◽  
Ring Extension ◽  
Power Series Ring ◽  
Flat Extension ◽  
Series Ring

Let R be a commutative ring with identity, and let S be a (not necessarily saturated) multiplicative subset of R. We define R to be a nonnil-S-Noetherian ring if each nonnil ideal of R is S-finite. In this paper, we study some properties of nonnil-S-Noetherian rings. More precisely, we investigate nonnil-S-Noetherian rings via the Cohen-type theorem, the flat extension, the faithfully flat extension, the polynomial ring extension, and the power series ring extension.

scholarly journals Exactness, Tor and Flat Modules Over a Commutative Ring

2004 ◽  
Vol 3 (3) ◽  
Author(s):  
Abhishek Banerjee
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Necessary Condition ◽  
Finitely Generated ◽  
Injective Modules ◽  
Flat Modules ◽  
Special Cases ◽  
Derived Functors ◽  
Finitely Generated Modules

In this paper, we principally explore flat modules over a commutative ring with identity. We do this in relation to projective and injective modules with the help of derived functors like Tor and Ext. We also consider an extension of the property of flatness and induce analogies with the “special cases” occurring in flat modules. We obtain some results on flatness in the context of a noetherian ring. We also characterize flat modules generated by one element and obtain a necessary condition for flatness of finitely generated modules.

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