s-Sequences and Monomial Modules

In this paper we study a monomial module M generated by an s-sequence and the main algebraic and homological invariants of the symmetric algebra of M. We show that the first syzygy module of a finitely generated module M, over any commutative Noetherian ring with unit, has a specific initial module with respect to an admissible order, provided M is generated by an s-sequence. Significant examples complement the results.

On the primefulness of local cohomology modules

Let be a commutative Noetherian ring with identity For a non-zero module . We prove that a multiplication primeful module and are I-cofinite and primeful, for each where is an ideal of with . As a consequence, we deduce that, if and are multiplication primeful R- modules, then is primeful. Another result is, for a projective module over an integral domain, admits projective resolution such that each is primeful (faithfully flat).

On strongly coseparable modules

A module M is called $$\mathfrak {s}$$ s -coseparable if for every nonzero submodule U of M such that M/U is finitely generated, there exists a nonzero direct summand V of M such that $$V \subseteq U$$ V ⊆ U and M/V is finitely generated. It is shown that every non-finitely generated free module is $$\mathfrak {s}$$ s -coseparable but a finitely generated free module is not, in general, $$\mathfrak {s}$$ s -coseparable. We prove that the class of $$\mathfrak {s}$$ s -coseparable modules over a right noetherian ring is closed under finite direct sums. We show that the class of commutative rings R for which every cyclic R-module is $$\mathfrak {s}$$ s -coseparable is exactly that of von Neumann regular rings. Some examples of modules M for which every direct summand of M is $$\mathfrak {s}$$ s -coseparable are provided.

ON INTERMEDIATE RINGS WHICH ARE FINITELY GENERATED MODULES OVER A NOETHERIAN RING

Cho (R, m) là vành giao hoán Noether và Q(R) là vành các thương toàn phần của R. Mục đích của bài báo này là nghiên cứu cấu trúc của các vành trung gian giữa R và Q(R). Gọi X là tập tất cả các lớp tương đương [I], trong đó I là ideal của R sao cho I 2 = aI với a ∈ I là phần tử không là ước của không trong R. Gọi Y là tập tất cả các vành trung gian A giữa R và Q(R) sao cho A là R-môđun hữu hạn sinh. Trong bài báo này, chúng tôi thiết lập một song ánh từ X đến Y. Một số ví dụ được đưa ra để làm rõ kết quả. Thứ nhất, chúng tôi chỉ ra nếu R là một miền ideal chính thì R là phần tử duy nhất của Y. Thứ hai, cho một vành Buchsbaum R mà không là Cohen-Macaulay, chúng tôi xây dựng một vành trung gian Cohen-Macaulay A ∈ Y. Để giải quyết vấn đề, chúng tôi áp dụng phương pháp nghiên cứu của S. Goto năm 1983, L. T. Nhàn và M. Brodmann 2012.

Notes on endomorphisms, local cohomology and completion

Let M M denote a finitely generated module over a Noetherian ring R R . For an ideal I ⊂ R I \subset R there is a study of the endomorphisms of the local cohomology module H I g ( M ) , g = g r a d e ( I , M ) , H^g_I(M), g = grade(I,M), and related results. Another subject is the study of left derived functors of the I I -adic completion Λ i I ( H I g ( M ) ) \Lambda ^I_i(H^g_I(M)) , motivated by a characterization of Gorenstein rings given in [25]. This provides another Cohen-Macaulay criterion. The results are illustrated by several examples. There is also an extension to the case of homomorphisms of two different local cohomology modules.

Cardinalities of prime spectra of precompletions

Given a complete local (Noetherian) ring T T , we find necessary and sufficient conditions on T T such that there exists a local domain A A with | A | > | T | |A| > |T| and A ^ = T \widehat {A} = T , where A ^ \widehat {A} denotes the completion of A A with respect to its maximal ideal. We then find necessary and sufficient conditions on T T such that there exists a domain A A with A ^ = T \widehat {A} = T and | S p e c ( A ) | > | S p e c ( T ) | |\mathrm {Spec}(A)| > |\mathrm {Spec}(T)| . Finally, we use “partial completions” to create local rings A A with A ^ = T \widehat {A} = T such that S p e c ( A ) \mathrm {Spec}(A) has varying cardinality in different varieties.

Dimension of finite free complexes over commutative Noetherian rings

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.

Filtrations in Module Categories, Derived Categories, and Prime Spectra

Let $R$ be a commutative noetherian ring. The notion of $n$-wide subcategories of ${\operatorname{\mathsf{Mod}}}\ R$ is introduced and studied in Matsui–Nam–Takahashi–Tri–Yen in relation to the cohomological dimension of a specialization-closed subset of ${\operatorname{Spec}}\ R$. In this paper, we introduce the notions of $n$-coherent subsets of ${\operatorname{Spec}}\ R$ and $n$-uniform subcategories of $\mathsf{D}({\operatorname{\mathsf{Mod}}}\ R)$ and explore their interactions with $n$-wide subcategories of ${\operatorname{\mathsf{Mod}}}\ R$. We obtain a commutative diagram that yields filtrations of subcategories of ${\operatorname{\mathsf{Mod}}}\ R$, $\mathsf{D}({\operatorname{\mathsf{Mod}}}\ R)$ and subsets of ${\operatorname{Spec}}\ R$ and complements classification theorems of subcategories due to Gabriel, Krause, Neeman, Takahashi, and Angeleri Hügel–Marks–Šťovíček–Takahashi–Vitória.

Vanishing and Artinianness of graded generalized local cohomology

UDC 512.5 Let be a homogeneous Noetherian ring with semilocal base ring Let be the irrelevant ideal of For two finitely generated graded -modules and several results on the vanishing, Artiniannes and tameness property of the graded -modules will be investigated.