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Let R be a commutative noetherian ring, I an ideal of R , and M a finitely generated R -module. We consider the asymptotic injective dimensions, projective dimensions, Bass numbers, and Betti numbers of localizations of M / I n M at prime ideals of R and prove that these invariants are stable or have polynomial growth for large integers n that do not depend on the prime ideals. [ABSTRACT FROM AUTHOR]

Let (A , m) be a Noetherian local ring and let M be a finitely generated Cohen Macaulay A module. Let G (A) = ⨁ n ≥ 0 m n / m n + 1 be the associated graded ring of A and G (M) = ⨁ n ≥ 0 m n M / m n + 1 M be the associated graded module of M. If A is regular and if G (M) has a quasi-pure resolution then we show that G (M) is Cohen-Macaulay. If A is Gorenstein and G (A) is Cohen-Macaulay and if M has finite projective dimension then we give lower bounds on e 0 (M) and e 1 (M). Finally let A = Q / (f 1 , ... , f c) be a strict complete intersection with ord (f i) = s for all i , where Q is a regular local ring. Let M be an Cohen-Macaulay module with complexity cx A (M) = r < c. We give lower bounds on e 0 (M) and e 1 (M). [ABSTRACT FROM AUTHOR]

Let A$A$ be a Noetherian domain and R$R$ be a finitely generated A$A$‐algebra. We study several features regarding the generic freeness over A$A$ of an R$R$‐module. For an ideal I⊂R$I \subset R$, we show that the local cohomology modules HIi(R)$\normalfont \text{H}_I^i(R)$ are generically free over A$A$ under certain settings where R$R$ is a smooth A$A$‐algebra. By utilizing the theory of Gröbner bases over arbitrary Noetherian rings, we provide an effective method to b make explicit the generic freeness over A$A$ of a finitely generated R$R$‐module. [ABSTRACT FROM AUTHOR]

We study classes of modules closed under direct sums, ℳ-submodules and ℳ-epimorphic images where ℳ is either the class of embeddings, RD-embeddings or pure embeddings.We show that the ℳ-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and 픰-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah’s non-forking to abstract elementary classes.We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity, S ⊆ R be a multiplicative set, and M be an R-module. We say that a submodule N of M with (N : R M) ∩ S = ∅ has an S-primary decomposition if it can be written as a finite intersection of S-primary submodules of M. In this paper, first we provide an example of the S-Noetherian module in which a submodule does not have a primary decomposition. Then our main aim of this paper is to establish the existence and uniqueness of S-primary decomposition in S-Noetherian modules as an extension of a classical Lasker-Noether primary decomposition theorem for Noetherian modules. [ABSTRACT FROM AUTHOR]

Let a be an ideal of a given commutative Noetherian ring R which satisfies the condition of the first problem of R. Hartshorne in [Affine duality and cofiniteness, Invent. Math. 9 (1970), 145–164]. In this paper, we prove that a also satisfies the condition of his second problem in the same article. We also provide an example to show that the converse statement does not hold in general. [ABSTRACT FROM AUTHOR]

In this survey, we study the defining ideals 픭 of the space monomial curves (ta,tb,tc) for primitive Pythagorean triples (PPT) a,b,c. We give a minimal system of generators of 픭. Hence, we provide the formula for Frobenius number of a PPT. Additionally, we show that over every field of arbitrary characteristic, there are distinct primitive Pythagorean triples such that their symbolic Rees algebras are noetherian. [ABSTRACT FROM AUTHOR]

Let A be a commutative Noetherian local ring with maximal ideal 픪, and let I be an ideal. The fiber cone is then an image of the polynomial ring over the residue field A/픪. The kernel of this map is called the defining ideal, and it is natural to ask how to compute it. In this paper, we provide a construction for the defining ideals of Cohen–Macaulay fiber cones. [ABSTRACT FROM AUTHOR]

Let S be a standard graded polynomial ring over a field K in a finite set of variables, and let 픪 be the graded maximal ideal of S. It is known that for a finitely generated graded S-module M and all integers k ≫ 0, the module 픪kM is componentwise linear. For large k we describe the pattern of the Betti table of 픪kM when depthM > 0. Moreover, we show that for any k ≫ 0, 픪kI has linear quotients if I is a monomial ideal. [ABSTRACT FROM AUTHOR]

Let (R, 픪) be a local commutative Noetherian ring and I an ideal of R. Assume that M is a finitely generated R-module of finite projective dimension p and N is a finitely generated R-module of dimension d. We prove that H픪1(H Ip+d−1(M,N)) is Artinian. Moreover, if HomR(R/I,HIp+d−1(M,N)) is finitely generated, then HIp+d−1(M,N) is an I-cofinite R-module. [ABSTRACT FROM AUTHOR]

Let (A , m A) be a Cohen–Macaulay local ring. Let M be a finitely generated A -module and let M ∗ denote the A -dual of M. Furthermore, if M ∗ is a maximal Cohen–Macaulay A -module, then we prove that μ A (M ∗) ≤ μ A (M) e (A) , where μ A (M) is the cardinality of a minimal generating set of M as an A -module and e (A) is the multiplicity of the local ring A. Furthermore, if M is a reflexive A -module then μ A (M) e (A) ≤ μ A (M ∗). As an application, we study the bound on the minimal number of generators of specific modules over two-dimensional normal local rings. We also mention some relevant examples. [ABSTRACT FROM AUTHOR]

Let ( S , 픫 ) {(S,{\mathfrak{n}})} be a commutative noetherian local ring and let ω ∈ 픫 {\omega\in{\mathfrak{n}}} be non-zerodivisor. This paper is concerned with the two categories of monomorphisms between finitely generated (Gorenstein) projective S-modules, such that their cokernels are annihilated by ω. It is shown that these categories, which will be denoted by 햬허헇 ⁢ ( ω , 풫 ) {{\mathsf{Mon}}(\omega,\mathcal{P})} and 햬허헇 ⁢ ( ω , 풢 ) {{\mathsf{Mon}}(\omega,\mathcal{G})} , are both Frobenius categories with the same projective objects. It is also proved that the stable category 햬허헇 ¯ ⁢ ( ω , 풫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} is triangle equivalent to the category of D-branes of type B, 햣햡 ⁢ ( ω ) {\mathsf{DB}(\omega)} , which has been introduced by Kontsevich and studied by Orlov. Moreover, it will be observed that the stable categories 햬허헇 ¯ ⁢ ( ω , 풫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} and 햬허헇 ¯ ⁢ ( ω , 풢 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{G})} are closely related to the singularity category of the factor ring R = S / ( ω ) {R=S/({\omega)}} . Precisely, there is a fully faithful triangle functor from the stable category 햬허헇 ¯ ⁢ ( ω , 풢 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{G})} to 햣 헌헀 ⁡ ( R ) {\operatorname{\mathsf{D_{sg}}}(R)} , which is dense if and only if R (and so S) are Gorenstein rings. Particularly, it is proved that the density of the restriction of this functor to 햬허헇 ¯ ⁢ ( ω , 풫 ) {\underline{\mathsf{Mon}}(\omega,\mathcal{P})} , guarantees the regularity of the ring S. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring and let M be an R-module. In this note, we give a brief proof of the Hilbert basis theorem for Noetherian modules. This states that if R contains the identity and M is a Noetherian unitary R-module, then M[X] is a Noetherian R[X]-module. We also show that if M[X] is a Noetherian R[X]-module, then M is a Noetherian R--module and there exists an element e ∈ R such that em = m for all m ∈ M. Finally, we prove that if M[X] is a Noetherian R[X]-module and annR(M) = (0), then R has the identity and M is a unitary R-module. [ABSTRACT FROM AUTHOR]

In the derived category of a commutative noetherian ring, we explicitly construct a silting object associated with each sp-filtration of the Zariski spectrum satisfying the "slice" condition. Our new construction is based on local cohomology and it allows us to study when the silting object is tilting. For a ring admitting a dualizing complex, this occurs precisely when the sp-filtration arises from a codimension function on the spectrum. In the absence of a dualizing complex, the situation is more delicate and the tilting property is closely related to the condition that the ring is a homomorphic image of a Cohen–Macaulay ring. We also provide dual versions of our results in the cosilting case. [ABSTRACT FROM AUTHOR]

In this paper, we will introduce and study the homological dimensions defined in the context of commutative rings with prime nilradical. So all rings considered in this paper are commutative with identity and with prime nilradical. We will introduce a new class of modules which are called ϕ -u-projective which generalizes the projectivity in the classical case and which is different from those introduced by the authors of [Y. Pu, M. Wang and W. Zhao, On nonnil-commutative diagrams and nonnil-projective modules, Commun. Algebra, doi:10.1080/00927872.2021.2021223; W. Zhao, On ϕ -exact sequence and ϕ -projective module, J. Korean Math. 58(6) (2021) 1513–1528]. Using the notion of ϕ -flatness introduced and studied by the authors of [G. H. Tang, F. G. Wang and W. Zhao, On ϕ -Von Neumann regular rings, J. Korean Math. Soc. 50(1) (2013) 219–229] and the nonnil-injectivity studied by the authors of [W. Qi and X. L. Zhang, Some Remarks on ϕ -Dedekind rings and ϕ -Prüfer rings, preprint (2022), arXiv:2103.08278v2 [math.AC]; X. Y. Yang, Generalized Noetherian Property of Rings and Modules (Northwest Normal University Library, Lanzhou, 2006); X. L. Zhang, Strongly ϕ -flat modules, strongly nonnil-injective modules and their homological dimensions, preprint (2022), https://arxiv.org/abs/2211.14681; X. L. Zhang and W. Zhao, On Nonnil-injective modules, J. Sichuan Normal Univ. 42(6) (2009) 808–815; W. Zhao, Homological theory over NP-rings and its applications (Sichuan Normal University, Chengdu, 2013)], we will introduce the ϕ -injective dimension, ϕ -projective dimension and ϕ -flat dimension for modules, and also the ϕ -(weak) global dimension of rings. Then, using these dimensions, we characterize several ϕ -rings (ϕ -Prüfer, ϕ -chained, ϕ -von Neumann, etc). Finally, we study the ϕ -(weak) global dimension of the trivial ring extensions defined by some conditions. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and S be a multiplicative subset of R. We say that R has S -Noetherian spectrum if for every ideal I of R , s I ⊆ J ⊆ I for some s ∈ S and some finitely generated ideal J. In this paper, we study rings with S -Noetherian spectrum. Among other things, we give a necessary and sufficient condition for Nagata's idealization R (+) M , where M is an R -module, to satisfy the S -Noetherian spectrum. We also investigate when the amalgamated algebra along an ideal has S -Noetherian spectrum. Several examples are given to illustrate the concepts and results. [ABSTRACT FROM AUTHOR]

Let s , a , n , d be positive integers such that n ≥ 2 and GCD (a , d) = 1 . Let P denote the defining ideal of the monomial curve associated to the sequence a , sa + d , ... , sa + nd . In this survey, we investigate the symbolic powers of P. We first show that for each i > 1 the equality P (i) = P i holds if and only if either n = 2 and a even or n = 3 , i = 2 and a ≡ 2 (mod 3) . The finitely generated property of the symbolic Rees algebra R S (P) are also explored. [ABSTRACT FROM AUTHOR]

Let R be an associative unital ring, and let f ∈ R [ x ] . We say that f is a left annihilating content (AC) polynomial if f = af1 for some a ∈ R and f 1 ∈ R [ x ] with l R [ x ] (f 1) = 0 . The ring R is called a left EM-ring if each f ∈ R [ x ] is a left AC polynomial. In this paper, it is shown that R is a left EM-ring if and only if R is a left McCoy ring, and for each finitely generated right ideal I of R, there is an element a ∈ R and a finitely generated right ideal J of R with l R (J) = 0 and I = aJ. If R is a left duo right Bezout ring, then R is a left EM-ring and has property (A). For a unique product monoid G, we show that if R is a reversible left EM-ring, then the monoid ring R [ G ] is also a left EM-ring. Additionally, for a reversible right Noetherian ring R, we prove that R, R [ x ] , R [ x , x − 1 ] , and R [ [ x ] ] are all simultaneously left EM-rings. Finally, we give an application of left EM-rings (resp. strongly left EM-rings) in studying the graph of zero-divisors of polynomial rings (resp. power series rings). [ABSTRACT FROM AUTHOR]

In this paper, some properties of finitely generated two-periodic modules over commutative Noetherian local rings have been studied. We show that under certain assumptions on a pair of modules (M,N) with M being two-periodic, the natural map M ⊗RN →HomR(M∗,N) is an isomorphism. As a consequence, we prove that Auslander’s depth formula holds for such a Tor-independent pair. Tor-independence plays a crucial role for the depth formula to hold. Under certain assumptions on the modules, we show that a pair of modules, over a one-dimensional local ring, is Tor-independent if and only if their tensor product is torsion-free. Celikbas et al. recently showed the Huneke–Wiegand conjecture holds for two-periodic modules over one-dimensional domains. We generalize their result to the case of two-periodic modules with rank over one-dimensional local rings. [ABSTRACT FROM AUTHOR]

For any finitely generated module M with non-zero rank over a commutative one dimensional Noetherian local domain, the numerical invariant h (M) was introduced and studied in [25]. We establish a bound on it which helps capture information about the torsion submodule of M when M has rank one and generalizes the discussion in [25]. We further study bounds and properties of h (M) in the case when M is the canonical module ω R. This in turn helps in answering a question of S. Greco and then provides classifications in the Gorenstein, almost Gorenstein and far-flung Gorenstein setups. [ABSTRACT FROM AUTHOR]

Let x _ denote a system of elements of a commutative ring R. For an R -module M we investigate when x _ is M -pro-regular resp. M -weakly pro-regular as generalizations of M -regular sequences. This is done in terms of Čech co-homology resp. homology, defined by H i ( C ˇ x _ ⊗ R ⋅) resp. by H i (R Hom R ( C ˇ x _ , ⋅)) ≅ H i (Hom R (L x _ , ⋅)) , where C ˇ x _ denotes the Čech complex and L x _ is a bounded free resolution of it as constructed in [17] resp. [16]. The property of x _ being M -pro-regular resp. M -weakly pro-regular follows by the vanishing of certain Čech co-homology resp. homology modules, which is related to completions. This extends previously work by Greenlees and May (see [5]) and Lipman et al. (see [1]). This contributes to a further understanding of Čech (co-)homology in the non-Noetherian case. As a technical tool we use one of Emmanouil's results (see [4]) about the inverse limits and its derived functor. As an application we prove a global variant of the results with an application to prisms in the sense of Bhatt and Scholze (see [3]). [ABSTRACT FROM AUTHOR]

AbstractIn this note, we introduce a very crude but natural notion of measure on the class of left R-modules over a ring R. We use this notion to give short proofs of some classical theorems on (left) Artinian rings and modules, due to Akizuki, Anderson, Hopkins, and Levitzki, as well as of some new results. [ABSTRACT FROM AUTHOR]

AbstractWe prove that if f:R→S is a local homomorphism of noetherian local rings, and M is a non-zero finitely generated or artinian S-module whose injective dimension over R is bounded by the difference of the embedding dimensions of R and S, then M is an injective S-module and f is an exceptional complete intersection map. [ABSTRACT FROM AUTHOR]

Let A be a ring and λ : R ≥ 0 → R ≥ 0 be an increasing nonzero function. In this paper, we show that if A is a Noetherian ring with characteristic n ≠ 0 and lim k → + ∞ λ (k + 1) λ (k) = + ∞ , then A [ X , Y ; λ ] is an SFT ring. This result allows us to construct a nonNoetherian SFT ring which has some Noetherian completion. Also, we use the composite ring extension to construct many examples of such rings. Many examples are provided. [ABSTRACT FROM AUTHOR]

Let φ : R → S be a flat local homomorphism between commutative Noetherian local rings. In this paper, the ascent and descent of Artinian module structures between R and S are investigated. For an Artinian R-module A, the structure of A ⊗ R S is described. As an application, the Artinianess of certain local cohomology modules is clarified. [ABSTRACT FROM AUTHOR]

Let 픞 be an ideal of a commutative noetherian ring R and M,N two R-modules with M finitely generated. It is shown that if either H픞i(N) is an 픞-cofinite module of dimension ≤ 1 for all i, or 픞 is a principal ideal and ExtRi(R/픞,N) is finitely generated for all i, or ExtRi(R/픞,N) is finitely generated and dimRH픞i(M,N) ≤ 1 for all i, then the R-module H픞t(M,N) is 픞-cofinite for all t ≥ 0. [ABSTRACT FROM AUTHOR]

Let (R, 픪) be a unitary commutative local Noetherian ring and I ⊂ R be an ideal. The aim of this paper is twofold: In the first part of this paper, we consider locally cohomologically complete intersection ideals of pure height dim R − 2. Next, we deal with cofinite modules. In particular, we investigate conditions for cofiniteness of HIh(R),h =ht(I) and conditions under which, the modules ExtRi(N,H픪dim R(R)) are of finite length for all i > 0 and all finitely generated R-module N. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring. In this paper, we find out close relationships between the module M being embedded in a module of projective dimension at most n and the (n + 1) -torsionfreeness of the n th syzygy of M. As an application, when R is local with residue field k , we compute the dimensions as k -vector spaces of Ext modules related to syzygies of k. [ABSTRACT FROM AUTHOR]

AbstractGiven a Noetherian ring A, the collection of integrally closed ideals in A which contain a nonzerodivisor forms a cancellative monoid under the operation I*J=IJ¯ , the integral closure of the product. The monoid is torsion-free and atomic. Restricting to monomial ideals in a polynomial ring, there is a surjective homomorphism from the Integral Polytope Group onto the Grothendieck group of integrally closed monomial ideals under translation invariance of their Newton Polyhedra. The Integral Polytope Group, the Grothendieck group of polytopes with integer vertices under Minkowski addition and translation invariance, has an explicit basis, allowing for explicit factoring in the monoid. [ABSTRACT FROM AUTHOR]

A right module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. We show the closures properties for certain high submodules by the QTAG-modules and vice versa. Important generalizations and certain related assertions of classical results in this direction are also established. [ABSTRACT FROM AUTHOR]

Our main goal in this note is to use a version of a lemma by Hanes and Huneke to provide characterizations of when certain one-dimensional reduced local rings are regular. This is of interest in view of the long-standing Berger's Conjecture (the ring is predicted to be regular if its universally finite differential module is torsion-free), which in fact we show to hold under suitable additional conditions, mostly toward the G-regular case of the conjecture. Furthermore, applying the same lemma to a Cohen-Macaulay local ring which is locally Gorenstein on the punctured spectrum but of arbitrary dimension, we notice a numerical characterization of when an ideal is strongly non-obstructed and of when a given semidualizing module is free. [ABSTRACT FROM AUTHOR]

Let G be a reductive group over a non-archimedean local field F of residue characteristic p. We prove that the Hecke algebras of G(F), with coefficients in any noetherian \mathbb {Z}_{\ell }-algebra R with \ell \neq p, are finitely generated modules over their centers, and that these centers are finitely generated R-algebras. Following Bernstein's original strategy, we then deduce that "second adjointness" holds for smooth representations of G(F) with coefficients in any \mathbb {Z}[\frac {1}{p}]-algebra. These results had been conjectured for a long time. The crucial new tool that unlocks the problem is the Fargues-Scholze morphism between a certain "excursion algebra" defined on the Langlands parameters side and the Bernstein center of G(F). Using this bridge, our main results are representation theoretic counterparts of the finiteness of certain morphisms between coarse moduli spaces of local Langlands parameters that we also prove here, which may be of independent interest. [ABSTRACT FROM AUTHOR]

Over the years questions have arisen about T-ideals of (noncommutative) polynomials. But when evaluating a noncentral polynomial in subalgebras of matrices, one often has little control in determining the specific evaluations of the polynomial. One way of overcoming this difficulty in characteristic 0, is to reduce to multilinear polynomials and to utilize the representation theory of the symmetric group. But this technique is unavailable in characteristic p > 0. An alternative method, which succeeds, is the process of "hiking" a polynomial, in which one specializes its indeterminates in several stages, to obtain a polynomial in which Capelli polynomial is embedded, in order to get control on its evaluations. This method was utilized on homogeneous polynomials in the proof of Specht's conjecture for affine algebras over fields of positive characteristic. In this paper, we develop hiking further to nonhomogeneous polynomials, to apply to the "representability question." Kemer proved in 1988 that every affine relatively free PI algebra over an infinite field, is representable. In 2010, the first author of this paper proved more generally that every affine relatively free PI algebra over any commutative Noetherian unital ring is representable [A. Belov, Local finite basis property and local representability of varieties of associative rings, Izv. Russian Acad. Sci. (1) (2010) 3–134. English Translation Izv. Math. 74(1) (2010) 1–126]. We present a different, complete, proof, based on hiking nonhomogeneous polynomials, over finite fields. We then obtain the full result over a Noetherian commutative ring, using Noetherian induction on T-ideals. The bulk of the proof is for the case of a base field of positive characteristic. Here, whereas the usage of hiking is more direct than in proving Specht's conjecture, one must consider nonhomogeneous polynomials when the base ring is finite, which entails certain difficulties to be overcome. In Appendix A, we show how hiking can be adapted to prove the involutory versions, as well as various graded and nonassociative theorems. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with identity and M an R -module. We say that M satisfies strong accr ∗ if for every submodule N of M and for every sequence 〈 r n 〉 of elements of R , the ascending sequence of submodules of the form, N : M r 1 ⊆ N : M r 1 r 2 ⊆ N : M r 1 r 2 r 3 ⊆ ⋯ is stationary. We say that a ring R satisfies strong accr ∗ if R regarded as a module over R satisfies strong accr ∗. In this paper, we give a necessary and sufficient condition for the pulback (respectively, the Nagata's idealization R (+) M) to be strong accr ∗ ring. We also prove a new characterization for a valuation ring to be strong accr ∗. [ABSTRACT FROM AUTHOR]

Let R bearing and n a fixed positive integer. A right R-moduleM is called strongly J-n-injective if every R-homomorphism from an n-generated small submodule of a free right R-module F to M extends to a homomorphism of F to M; a right R-module V is said to be strongly J-n-flat, if for every ngenerated small submodule T of a free left R-module F, the canonical map V ⊗ T → V ⊗ F is monic; a ring R is called left strongly J-n-coherent if every n-generated small submodule of a free left R-module is finitely presented; a ring R is said to be left J-n-semihereditary if every n-generated small left ideal of R is projective. We study strongly J-n-injective modules, strongly J-n-flat modules and left strongly J-n-coherent rings. Using the concepts of strongly J-n-injectivity and strongly J-n-flatness of modules, we also present some characterizations of strongly J-n-coherent rings and J-n-semihereditary rings. [ABSTRACT FROM AUTHOR]

A module M over a ring is called simple-separable if every simple submodule of M is contained in a finitely generated direct summand of M. While a direct sum of any family of simple-separable modules is shown to be always simple-separable, we prove that a direct summand of a simple-separable module does not inherit the property, in general. It is also shown that an injective module M over a right noetherian ring is simple-separable if and only if M = M1 ⊕ M2 such that M1 is separable and M2 has zero socle. The structure of simple-separable abelian groups is completely described. [ABSTRACT FROM AUTHOR]

Let M = ⊕n∈ZMn be a strongly graded module over strongly graded ring D = ⊕n∈ZDn. In this paper, we prove that if M0 is a unique factorization module (UFM for short) over D0 and D is a unique factorization domain (UFD for short), then M is a UFM over D. Furthermore, if D0 is a Noetherian domain, we give a necessary and sufficient condition for a positively graded module L = ⊕n∈Z0Mn to be a UFM over positively graded domain R = ⊕n∈Z0Dn. [ABSTRACT FROM AUTHOR]

There are different mechanisms in the literature to measure how flat a module can be. We study an alternative perspective on the analysis of the flatness of a module, as we assign to every module a class of short exact sequences of modules, namely flatly generated proper classes. We focus on modules that generate flat proper classes, aiming for them to be as small as possible. We refer to such modules as being τ-rugged, as opposed to flat modules. Properties of τ-rugged modules are studied. We study the structure of rings whose certain types of modules are either flat or τ-rugged. Specifically, we prove that if R is a right Noetherian ring, then every (finitely presented) nonflat right R-module is τ-rugged if and only if R has a unique (up to isomorphism) singular simple right R-module and it is either Artinian serial ring with J2(R) = 0 or right finitely ∑-CS, right SI ring. If R is a right perfect ring with nonflat finitely presented simple right R-module U, then U is τ-rugged if and only if every nonflat right R-module is τ-rugged if and only if R has a unique (up to isomorphism) singular simple right R-module U and it is Artinian serial ring with J2(R) = 0. In addition, if R is commutative and every nonflat R-module is τ-rugged, then R is either a von Neumann regular ring or an fp-injective ring. [ABSTRACT FROM AUTHOR]

This work presents results on the finiteness, and on the symmetry properties, of various homological dimensions associated to the Jacobson radical and its higher syzygies, of a semiperfect ring. [ABSTRACT FROM AUTHOR]

We develop a formalism of unit |$F$| -modules in the style of Lyubeznik and Emerton-Kisin for rings that have finite |$F$| -representation type after localization and completion at every prime ideal. As applications, we show that if |$R$| is such a ring then the iterated local cohomology modules |$H^{n_{1}}_{I_{1}} \circ \cdots \circ H^{n_{s}}_{I_{s}}(R)$| have finitely many associated primes, and that all local cohomology modules |$H^{n}_{I}(R / gR)$| have closed support when |$g$| is a nonzerodivisor on |$R$|⁠. [ABSTRACT FROM AUTHOR]

Over the years, prime submodules and their generalizations have played a pivotal role in commutative algebra, garnering considerable attention from numerous researchers and scholars in the field. This papers presents a generalization of 1-absorbing primary ideals, namely the classical 1-absorbing primary submodules. Let ℜ be a commutative ring and M an ℜ-module. A proper submodule K of M is called a classical 1-absorbing primary submodule of M, if x y z η ∈ K for some η ∈ M and nonunits x , y , z ∈ ℜ , then x y η ∈ K or z t η ∈ K for some t ≥ 1 . In addition to providing various characterizations of classical 1-absorbing primary submodules, we examine relationships between classical 1-absorbing primary submodules and 1-absorbing primary submodules. We also explore the properties of classical 1-absorbing primary submodules under homomorphism in factor modules, the localization modules and Cartesian product of modules. Finally, we investigate this class of submodules in amalgamated duplication of modules. [ABSTRACT FROM AUTHOR]

For a commutative ring R and a Hopf algebra H which is finitely generated projective as an R-module, it is established that there is an (anti)-isomorphism of groups between the Brauer group BQ(R, H) of Hopf Yetter-Drinfel'd H-module algebras and the Brauer group BQ (R , H *) of Hopf Yetter-Drinfel'd H * -module algebras, where H * is the linear dual of H. In this paper, we generalize this result by establishing an anti-isomorphism of groups between BQ(S, H), the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S, H)-module algebras and BQ (S o p , H *) , the Brauer group of dyslectic Hopf Yetter-Drinfel'd (S o p , H *) -module algebras, where S is an H-commutative Hopf Yetter-Drinfel'd H-module algebra and Sop is the opposite algebra of S. Communicated by Alberto Elduque [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring and M be an R-module. The R-module M is called distributive if for every submodules S, T and U of M, the equality S ∩ (T + U) = S ∩ T + S ∩ U holds true. In this paper, we give a necessary and sufficient condition for M to be distributive based on injective envelopes. The proof uses Matlis' results on injective modules. [ABSTRACT FROM AUTHOR]

Let a be an ideal of Noetherian ring R and M a finitely generated R-module such that cd (a , M) = c . In this paper, we investigate Att R (H a c (M)) . Among other things, it is shown that Max { p ∈ Supp R M | cd (a , R / p) = c } ⊆ Att R (H a c (M)) . We also show that Att R (H a c (M)) = { p ∈ Supp R M | cd (a , R / p) = c , p = Ann R (H a c (R / p)) } and { p ∈ Supp R M | cd (a , R / p) = c , dim R / p − 1 ≤ cd (a , R / p) ≤ dim R / p } ⊆ Att R (H a c (M)). Finally, we prove that if (R , m) is a local ring and dim R / a = 1 then Att R (H a c (M)) = { p ∈ Supp R M | cd (a , R / p) = cd (a , M) } . Then by using this, it is shown that if (R , m) is a local ring then { p ∈ Supp R M | cd (a , R / p) = c , dim R / (a + p) = 1 } ⊆ Att R (H a c (M)). [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of weakly semiprime ideals and weakly n -systems in noncommutative rings. We establish the equivalence between an ideal P being a weakly semiprime ideal and R − P being a weakly n -system. We provide alternative definitions and explore the properties of weakly semiprime ideals. Additionally, we investigate scenarios where all ideals in a given ring are weakly semiprime and demonstrate that in Noetherian rings, where every ideal is weakly semiprime, the prime radical and the Jacobson radical coincide. [ABSTRACT FROM AUTHOR]

This paper is a commutative algebra introduction to the homological theory of quasi-coherent sheaves and contraherent cosheaves over quasi-compact semi-separated schemes. Antilocality is an alternative way in which global properties are locally controlled in a finite affine open covering. For example, injectivity of modules over non-Noetherian commutative rings is not preserved by localizations, while homotopy injectivity of complexes of modules is not preserved by localizations even for Noetherian rings. The latter also applies to the contraadjustedness and cotorsion properties. All the mentioned properties of modules or complexes over commutative rings are actually antilocal. They are also colocal, if one presumes contraadjustedness. Generally, if the left class in a (hereditary complete) cotorsion theory for modules or complexes of modules over commutative rings is local and preserved by direct images with respect to affine open immersions, then the right class is antilocal. If the right class in a cotorsion theory for contraadjusted modules or complexes of contraadjusted modules is colocal and preserved by such direct images, then the left class is antilocal. As further examples, the class of flat contraadjusted modules is antilocal, and so are the classes of acyclic, Becker-coacyclic, or Becker-contraacyclic complexes of contraadjusted modules. The same applies to the classes of homotopy flat complexes of flat contraadjusted modules and acyclic complexes of flat contraadjusted modules with flat modules of cocycles. [ABSTRACT FROM AUTHOR]

Let R be a ring and S be a multiplicative subset of R. Then R is called a uniformly S-Noetherian ring if there exists s ∈ S such that, for any ideal I of R, sI ⊆ K for some finitely generated subideal K of I. We give the Eakin-Nagata-Formanek theorem for uniformly S-Noetherian rings. In addition, the uniformly S-Noetherian properties on several ring constructions are given. The notion of u-S-injective modules is also introduced and studied. Finally, we obtain the Bass-Papp theorem for uniformly S-Noetherian rings. [ABSTRACT FROM AUTHOR]

Given an additive category C and an integer n ≥ 2. The higher differential additive category consists of objects X in C equipped with an endomorphism ϵX satisfying ϵ X n = 0 . Let R be a finite-dimensional basic algebra over an algebraically closed field and T the augmenting functor from the category of finitely generated left R-modules to that of finitely generated left R/(tn)-modules. It is proved that a finitely generated left R-module M is τ-rigid (respectively, (support) τ-tilting, almost complete τ-tilting) if and only if so is T(M)as a left R[t]/(tn)-module. Moreover, R is τm-selfinjective if and only if so is R[t]/(tn). [ABSTRACT FROM AUTHOR]

We provide suitable conditions under which the asymptotic limit of the Hilbert-Samuel coefficients of the Frobenius powers of an \mathfrak {m}-primary ideal exists in a Noetherian local ring (R,\mathfrak {m}) with prime characteristic p>0. This, in turn, gives an expression of the Hilbert-Kunz multiplicity of powers of the ideal. We also prove that for a face ring R of a simplicial complex and an ideal J generated by pure powers of the variables, the generalized Hilbert-Kunz function \ell (R/(J^{[q]})^k) is a polynomial for all q,k and also give an expression of the generalized Hilbert-Kunz multiplicity of powers of J in terms of Hilbert-Samuel multiplicity of J. We conclude by giving a counter-example to a conjecture proposed by I. Smirnov which connects the stability of an ideal with the asymptotic limit of the first Hilbert coefficient of the Frobenius power of the ideal. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with 1 and M be a left unitary R-module. In this paper, we give a generalization for the notions of compressible (retractable) module. As well as, we study closed (St-closed) compressible and closed (St-closed) retractable. Furthermore, some of their advantages, properties, categorizations and instances have been given. Finally, we study the relation between them. [ABSTRACT FROM AUTHOR]