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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]

In this article we solve the following problem: "The Hilbert function of the local ring of a 4-generated pseudo-symmetric numerical semigroup is always non-decreasing." We give a complete characterization of the standard bases when the tangent cone is not Cohen-Macaulay by showing that the number of elements in the standard basis depends on some parameters we define. Since the tangent cone is not Cohen-Macaulay, non-decreasingness of the Hilbert function was not guaranteed, thus we proved the non-decreasingness from our explicit Hilbert function computation. [ABSTRACT FROM AUTHOR]

AbstractThe present paper deals with the investigation of the structure of general fiber product rings R×TS , where R, S and T are local rings with common residue field. We show that the Poincaré series of any R-module over the fiber product ring R×TS is bounded by a rational function. In addition, we give a description of depth(R×TS) , which is an open problem in this theory. As a biproduct, using the characterization of the Betti numbers over R×TS obtained, we provide certain cases of the Cohen-Macaulayness of R×TS and, in particular, we show that R×TS is always non-regular. Some positive answers for the Buchsbaum-Eisenbud-Horrocks and Total rank conjectures over R×TS are also established. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with identity. In this paper, formal expressions of the number of m × n matrices over R of rank r and the number of invertible matrices over R are presented. The number of matrices over R with a given rank and a given number of single unit-entry rows, rows in which a single entry is a unit and all other entries are zero, is finally determined. [ABSTRACT FROM AUTHOR]

Suppose R is a local ring with invariants p , n , r , m , k and m r = 4 , that is R of order p 4. Then, R = R 0 + u R 0 + v R 0 + w R 0 has maximal ideal J = u R 0 + v R 0 + w R 0 of order p (m − 1) r and a residue field F of order p r , where R 0 = G R (p n , r) is the coefficient subring of R. In this article, the goal is to improve the understanding of linear codes over small-order non-chain rings. In particular, we produce the MacWilliams formulas and generator matrices for linear codes of length N over R. In order to accomplish that, we first list all such rings up to isomorphism for different values of p , n , r , m , k. Furthermore, we fully describe the lattice of ideals in R and their orders. Next, for linear codes C over R, we compute the generator matrices and MacWilliams identities, as shown by numerical examples. Given that non-chain rings are not principal ideals rings, it is crucial to acknowledge the difficulties that arise while studying linear codes over them. [ABSTRACT FROM AUTHOR]

The notion of local rings with quasi-decomposable maximal ideal was formally introduced by Nasseh and Takahashi. In separate works, the authors of the present paper showed that such rings have rigid homological properties; for instance, they are both Ext- and Tor-friendly. One point of this paper is to further explore the homological properties of these rings and also introduce new classes of such rings from a combinatorial point of view. Another point is to investigate how far some of these homological properties can be pushed along certain diagrams of local ring homomorphisms. [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 (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]

We characterize which complete local (Noetherian) rings T containing the rationals are the completion of a countable excellent local ring S. We also discuss the possibilities for the map from the minimal prime ideals of T to the minimal prime ideals of S and we prove some characterization-style results. [ABSTRACT FROM AUTHOR]

Let R be a commutative ring with unity. The prime ideal sum graph of the ring R is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of R and two distinct vertices I, J are adjacent if and only if I + J is a prime ideal of R. In this paper, we characterize all commutative Artinian rings whose prime ideal sum graphs are line graphs. Finally, we give a description of all commutative Artinian rings whose prime ideal sum graph is the complement of a line graph. [ABSTRACT FROM AUTHOR]

In [D. Rees, Hilbert functions and pseudo-rational local rings of dimension two, J. London Math. Soc. (2) 24 (1981) 467–479], Rees gave a characterization for the normal joint reduction number zero of two 픪-primary ideals in an analytically unramified Cohen–Macaulay local ring of dimension two. Rees’ result is a generalization of Zariski’s product theorem for complete ideals in a regular local ring of dimension two. The aim of this paper is to extend Rees’ theorem for the ordinary powers of 픪-primary ideals I and J in a Cohen–Macaulay local ring of dimension two. Following Rees’ approach, we define the modified Koszul homology modules Mr,s1(ak,bk) for a joint reduction (a,b) of I and J. Under the additional assumption that the associated graded rings of I and J have positive depth, we obtain a characterization of the joint reduction number zero of I and J in terms of the vanishing of the module M0,01(a,b), as well as in terms of the Hilbert coefficients and the bigraded Hilbert coefficients. More generally, we introduce the joint reduction lattice and study the vanishing of Mr,s1(a,b) for any r,s ≥ 0. This gives a characterization for a vector (r,s) to be in the joint reduction lattice of I and J. We also give a cohomological interpretation of these theorems by investigating the local cohomology modules of the bigraded extended Rees algebra. This gives another characterization for a vector (r,s) to be in the joint reduction lattice and also extends a recent result of Masuti and Verma in [Local cohomology of bigraded Rees algebras and normal Hilbert coefficients, J. Pure Appl. Algebra 218(5) (2014) 904–918] for ordinary powers of ideals. [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 u , v , and w be indeterminates over F p m and let R = F p m + u F p m + v F p m + w F p m , where p is a prime. Then, R is a ring of order p 4 m , and R ≅ F p m [ u , v , w ] I with maximal ideal J = u F p m + v F p m + w F p m of order p 3 m and a residue field F p m of order p m , where I is an appropriate ideal. In this article, the goal is to improve the understanding of linear codes over local non-chain rings. In particular, we investigate the symmetrized weight enumerators and generator matrices of linear codes of length N over R. In order to accomplish that, we first list all such rings up to the isomorphism for different values of the index of nilpotency l of J, 2 ≤ l ≤ 4. Furthermore, we fully describe the lattice of ideals of R and their orders. Next, for linear codes C over R , we compute the generator matrices and symmetrized weight enumerators, as shown by numerical examples. [ABSTRACT FROM AUTHOR]

We generalize a theorem of Ding relating the generalized Loewy length g ℓ ℓ (R) and index of a one-dimensional Cohen-Macaulay local ring (R , m , k). Ding proved that if R is Gorenstein, the associated graded ring is Cohen-Macaulay, and k is infinite, then the generalized Loewy length and index of R are equal. However, if k is finite, equality may not hold. We prove that if the index of a one-dimensional Cohen-Macaulay local ring is finite and the associated graded ring has a homogeneous nonzerodivisor of degree t , then g ℓ ℓ (R) ≤ index (R) + t − 1. Next we prove that if R is a one-dimensional hypersurface ring with a witness to the generalized Loewy length that induces a regular initial form on the associated graded ring, then the generalized Loewy length achieves this upper bound. We then compute the generalized Loewy lengths of several families of examples of one-dimensional hypersurface rings over finite fields. Finally, we study a graded version of the generalized Loewy length and determine its value for numerical semigroup rings. [ABSTRACT FROM AUTHOR]

We prove that cellular Noetherian algebras with finite global dimension are split quasi-hereditary over a regular commutative Noetherian ring with finite Krull dimension and their quasi-hereditary structure is unique, up to equivalence. In the process, we establish that a split quasi-hereditary algebra is semi-perfect if and only if the ground ring is a local commutative Noetherian ring. We give a formula to determine the global dimension of a split quasi-hereditary algebra over a commutative regular Noetherian ring (with finite Krull dimension) in terms of the ground ring and finite-dimensional split quasi-hereditary algebras. For the general case, we give upper bounds for the finitistic dimension of split quasi-hereditary algebras over arbitrary commutative Noetherian rings. We apply these results to Schur algebras over regular Noetherian rings and to Schur algebras over quotients rings of the integers. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with nonzero identity. An ideal I of R is called nilpotent of index 2 if I 2 = 0. R is called local nilpotent of index 2 if it has a unique maximal nilpotent ideal of index 2. In this paper, we investigate the interplay between cliques in the zero divisor graph and nilpotent ideals of index 2 in R. In particular, if Ω is a maximal clique and Ω ∪ { 0 } is an ideal, then Ω ∪ { 0 } is a maximal nilpotent ideal of index 2. Also, we study local nilpotent rings of index 2. Finally, we prove that every local ring of order p n with n ≤ 3 is local nilpotent of index 2 and present a counter example for p 4 . [ABSTRACT FROM AUTHOR]

Let (A ,) be a Cohen–Macaulay local ring with infinite residue field A / and I be an -primary ideal of A. Let σ be a semi-prime operation in the set of ideals of A and σ (f I) = (σ (I n)) n ∈ ℕ be a I -good filtration on A. Let k ≥ 1 be an integer and (I k) σ = σ (I n + k) : A σ (I n) for all large integers n. In this paper, we give some relationships between (I k) σ and σ (f I) -superficial sequence. Thus, we show that if x 1 , x 2 , ... , x r is a maximal σ (f I) -superficial sequence on I and J = (x 1 , x 2 , ... , x r) , then it exists a positive integer n 0 such that J σ (I n) = σ (I n + 1) for all n ≥ n 0 . [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]

In this paper, we introduce the unit-Jacobson graph, which is defined by the unit elements and the elements of the Jacobson radical of a commutative ring R with nonzero identity. We give relationships between this new graph concept and some special rings such as j-clean rings, UJ-rings, local rings, and cartesian rings. Moreover, we investigate the concepts of the dominating set, diameter, and girth on the unit-Jacobson graph. [ABSTRACT FROM AUTHOR]

Suppose that R = Z p 4 [ u ] with u 2 = p 3 β and p u = 0 , where p is a prime and β is a unit in R. Then, R is a local non-chain ring of order p 5 with a unique maximal ideal J = (p , u) and a residue field of order p. A linear code C of length N over R is an R-submodule of R N. The purpose of this article is to examine MacWilliams identities and generator matrices for linear codes of length N over R. We first prove that when p ≠ 2 , there are precisely two distinct rings with these properties up to isomorphism. However, for p = 2 , only a single such ring is found. Furthermore, we fully describe the lattice of ideals of R and their orders. We then calculate the generator matrices and MacWilliams relations for the linear codes C over R , illustrated with numerical examples. It is important to address that there are challenges associated with working with linear codes over non-chain rings, as such rings are not principal ideal rings. [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]

A semidualizing module is a generalization of Grothendieck's dualizing module. For a local Cohen–Macaulay ring R, the ring itself and its canonical module are always realized as (trivial) semidualizing modules. Reasonably, one might ponder the question; when do nontrivial examples exist? In this paper, we study this question in the realm of numerical semigroup rings and, up to multiplicity 9, completely classify which of these rings possess a nontrivial semidualizing module. Using this classification, for each integer n > 8 , we construct a numerical semigroup ring of multiplicity n which admits a nontrivial semidualizing module. [ABSTRACT FROM AUTHOR]

We contribute to the theory of G-dimension relative to a semidualizing module C , in connection to the properties of a module being totally C -reflexive, C - k -torsionless, and C - k -syzygy, where k ≥ 0 is an integer. We extend several known results, and for an integer q ≥ 0 we introduce the class of C - q -Gorenstein rings. We also consider C -duals and initiate the study of C -valued derivation modules over a local ring R of low depth; if C = R and R is a three-dimensional Gorenstein domain, we find a bound for the number of generators and we propose a question when C is general. [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 R be a commutative ring with identity and Nil (R) be the ideal of all nilpotent elements of R. Let (R) = { I : I be a nontrivial ideal of R and there exists a nontrivial ideal J such that I J ⊆ Nil (R) }. The nil-graph of ideals of R is defined as the graph N (R) whose vertex set is the set (R) and two distinct vertices I and J are adjacent if and only if I J ⊆ Nil (R). A subset of vertices S ⊆ V (G) resolves a graph G , and S is a resolving set of G , if every vertex is uniquely determined by its vector of distances to the vertices of S. In particular, for an ordered subset S = { v 1 , v 2 , ... , v k } of vertices in a connected graph G and a vertex v ∈ V (G) \ S of G , the metric representation of v with respect to S is the k -vector D (v | S) = (d (v , v 1) , d (v , v 2) , ... , d (v , v k)). The set S is a resolving set for G if D (u | S) = D (v | S) implies that u = v , for all pair of vertices, v , u ∈ V (G) \ S. A resolving set S of minimum cardinality is the metric basis for G , and the number of elements in the resolving set of minimum cardinality is the metric dimension of G. If D (u | S) ≠ D (v | S) for every pair u , v of adjacent vertices of G , then S is called a local metric set of G. The minimum k for which G has a local metric k -set is the local metric dimension of G , which is denoted by lmd (G). In this paper, we determine metric dimension and local metric dimension of nil-graph of ideals of commutative rings. [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 p be a prime such that pm ≡ 1(mod4), where m is a positive integer. For any nonzero element α of 픽pm, we determine the algebraic structure of all α-constacyclic codes of length 4ps over the finite commutative chain ring 픽pm[u] 〈u3〉 ≅픽pm + u픽pm + u2픽 pm, where u3 = 0 and s is a positive integer. If the unit α ∈ 픽pm is a square, α = δ2, each α-constacyclic code of length 4ps is expressed as a direct sum of an − δ-constacyclic code and an δ- constacyclic code of length 2ps. In the main case that the unit α is not a square, it is shown that any nonzero polynomial of degree at most 3 over 픽pm is invertible in the ambient ring (픽pm+u픽pm+u2픽pm)[x] 〈x4ps−α〉. It is also proven that the ambient ring (픽pm+u픽pm+u2픽pm)[x] 〈x4ps−α〉 is a local ring with the unique maximal ideal 〈x4 − α 0,u〉, where α0ps = α. Such α-constacyclic codes are then classified into eight distinct types of ideals, and the detailed structures of ideals in each type are provided. Among other results, the number of codewords, and the dual of each α-constacyclic code are obtained. The non-existence of self-dual and isodual α-constacyclic codes of length 4ps over 픽pm + u픽pm + u2픽 pm, when the unit α is not a square, is likewise proved. [ABSTRACT FROM AUTHOR]

We establish new characterizations of complete intersection ideals in Cohen-Macaulay local rings as a byproduct of our investigation on a certain André-Quillen homology vector space in connection to the vanishing of (finitely many) Ext modules over a suitable quasi-deformation. Along the way, by means of (co)homology vanishing, we also detect criteria for quasi-complete intersection ideals. [ABSTRACT FROM AUTHOR]

We address the problem of when the tensor product of two finitely generated modules over a Cohen-Macaulay local ring is Ulrich in the generalized sense of Goto et al., and in particular in the original sense from the 80's. As applications, besides freeness criteria for modules, characterizations of complete intersections, and an Ulrich-based approach to the long-standing Berger's conjecture, we give simple proofs that two celebrated homological conjectures, namely the Huneke-Wiegand and the Auslander-Reiten problems, are true for the class of Ulrich modules. [ABSTRACT FROM AUTHOR]

This paper investigates the relation between the almost Gorenstein properties for graded rings and for local rings. Once R is an almost Gorenstein graded ring, the localization RM of R at the graded maximal ideal M is almost Gorenstein as a local ring. The converse does not hold true in general ([7, Theorems 2.7, 2.8], [8, Example 8.8]). However, it does for one-dimensional graded domains with mild conditions, which we clarify in the present paper. We explore the defining ideals of almost Gorenstein numerical semigroup rings as well. [ABSTRACT FROM AUTHOR]

We draw concrete consequences from our arithmetic duality for two-dimensional local rings with perfect residue field. These consequences include class field theory, Hasse principles for coverings and K2 and a duality between divisor class groups and Brauer groups. To obtain these, we analyze the ind-pro-algebraic group structures on arithmetic cohomology obtained earlier and prove some finiteness properties about them. [ABSTRACT FROM AUTHOR]

We study the concept of second module and extend it to more general environments. We also provide descriptions of simple left semiartinian, left local rings, semisimple and simple rings in terms of their A -second modules with respect to a preradical class. [ABSTRACT FROM AUTHOR]

We consider a local non-Frobenius ring R p = F p [ u , v ] / ⟨ u 2 , v 2 , u v , v u ⟩ defined for each prime number p and study the ideal representation of the ring R p [ x ] / ⟨ x n + 1 ⟩ , which is well-known to be related to negacyclic codes over R p of length n = p s with s ≥ 1 . We assume that n is a power of p, and under this setting, we show that a specific class of ideals can be represented uniquely in terms of degrees related to its generators. We also give a lower bound of the minimum Hamming distances of negacyclic codes over R p , and show that the lower bound is sharp for several codes whose corresponding ideals of R p [ x ] / ⟨ x n + 1 ⟩ belong to the aforementioned class. [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]

We introduce a new class of commutative noetherian DG-rings which generalizes the class of regular local rings. These are defined to be local DG-rings (A , m ¯) such that the maximal ideal m ¯ ⊆ H 0 (A) can be generated by an A -regular sequence. We call these DG-rings sequence-regular DG-rings, and make a detailed study of them. Using methods of Cohen-Macaulay differential graded algebra, we prove that the Auslander-Buchsbaum-Serre theorem about localization generalizes to this setting. This allows us to define global sequence-regular DG-rings, and to introduce this regularity condition to derived algebraic geometry. It is shown that these DG-rings share many properties of classical regular local rings, and in particular we are able to construct canonical residue DG-fields in this context. Finally, we show that sequence-regular DG-rings are ubiquitous, and in particular, any eventually coconnective derived algebraic variety over a perfect field is generically sequence-regular. [ABSTRACT FROM AUTHOR]

Let F be a field, let D be a local subring of F , and let Val F (D) be the space of valuation rings of F that dominate D. We lift Zariski's connectedness theorem for fibers of a projective morphism to the Zariski-Riemann space of valuation rings of F by proving that a subring R of F dominating D is local, residually algebraic over D and integrally closed in F if and only if there is a closed and connected subspace Z of Val F (D) such that R is the intersection of the rings in Z. Consequently, the intersection of the rings in any closed and connected subset of Val F (D) is a local ring. In proving this, we also prove a converse to Zariski's connectedness theorem. Our results do not require the rings involved to be Noetherian. [ABSTRACT FROM AUTHOR]

Let G be a smooth linear group scheme of finite type. For any positive integer k and a finite field F , let W k (F) be the ring of Witt vectors of length k over F. We show that the group algebras of G (F [ t ] / (t k)) and G (W k (F)) are isomorphic (i.e. the multi-sets of the dimensions of the irreducible representations are equal) for any positive integer k and finite field F with large enough characteristic. We also prove that if char F is large enough, then the cardinality of the set { dim ⁡ ρ | ρ ∈ irr (G (F)) } is bounded uniformly in F. [ABSTRACT FROM AUTHOR]

Let $({\cal{A}},{\cal{E}})$ be an exact category. We establish basic results that allow one to identify sub(bi)functors of ${\operatorname{Ext}}_{\cal{E}}(-,-)$ using additivity of numerical functions and restriction to subcategories. We also study a small number of these new functors over commutative local rings in detail and find a range of applications from detecting regularity to understanding Ulrich 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]

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]

The main objective of this article is to classify all finite singleton local rings, which are associative rings characterized by a unique maximal ideal and a distinguished basis consisting of a single element. These rings are associated with four positive integer invariants p , n , s , and t , where p is a prime number. In particular, we aim to classify these rings and count them up to isomorphism while maintaining the same set of invariants. We have found interesting cases of finite singleton local rings with orders of p 6 and p 7 that hold substantial importance in the field of coding theory. [ABSTRACT FROM AUTHOR]

This paper concerns the distribution of Selmer ranks in a family of even Galois representations in residual characteristic p = 2 obtained by allowing ramification at auxiliary primes. The main result is a Galois cohomological analogue of a theorem of Friedlander, Iwaniec, Mazur and Rubin on the distribution of Selmer ranks in a family of twists of elliptic curves. The Selmer groups are constructed as prescribed by the Galois cohomological method for GL (2) : At each ramified place, the local Selmer condition is the tangent space of a smooth quotient of the local deformation ring. By methods of global class field theory, the Selmer group at the minimal level is computed explicitly. The infinitude of primes for which the Selmer rank increases by one is proved, and the density of such primes is shown to be 1/192. The proof combines Wiles' formula and the global reciprocity law. The result has implications for the algebraic structure of even deformation rings and the distribution of their presentations in families. [ABSTRACT FROM AUTHOR]

Let B be a local (Noetherian) ring and suppose that B has n associated prime ideals where n ≥ 2 . We identify sufficient conditions for there to exist a local (Noetherian) subring S of B such that S and B have the same completion and S has exactly n − 1 associated prime ideals. We include applications and consequences of this result. [ABSTRACT FROM AUTHOR]

We call a real-valued function f on a topological space X a locally constant if each point x ∈ X has a neighborhood on which f is constant. The set of all locally constant functions f: X → ℝ is denoted by E(X). We observe that E(X) is the largest von Neumann regular subring of C(X) containing ℝ. We show that for every space X, there is a zero-dimensional space Y such that E(X) ≅ E(Y) and E(X) determines the topology of X if and only if X is a zero-dimensional space. By an example, it turns out that E(X) is not always of the form C(Y) for some space Y, but whenever X is a locally connected space, we observe that E(X) is a C(Y), where Y is a discrete space. Finally some von Neumann local subrings of C(X) containing ℝ are introduced and topological spaces X are characterized for which these subrings coincide with the smallest one. Note that a ring R is said to be a von Neumann local ring if for each a ∈ R either a or 1 − a has von Neumann inverse. [ABSTRACT FROM AUTHOR]

This article builds on recent work of the first three authors where a notion of congruence modules in higher codimension is introduced. The main results are a criterion for detecting regularity of local rings in terms of congruence modules, and a more refined version of a result tracking the change of congruence modules under deformation. Number theoretic applications include the construction of canonical lines in certain Galois cohomology groups arising from adjoint motives of Hilbert modular forms. [ABSTRACT FROM AUTHOR]

We prove that the multiplicity of a filtration of a local ring satisfies various convexity properties. In particular, we show the multiplicity is convex along geodesics. As a consequence, we prove that the volume of a valuation is log convex on simplices of quasi-monomial valuations and give a new proof of a theorem of Xu and Zhuang on the uniqueness of normalized volume minimizers. In another direction, we generalize a theorem of Rees on multiplicities of ideals to filtrations and characterize when the Minkowski inequality for filtrations is an equality under mild assumptions. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic 0, let S be a complete local ring with coefficient field K , let K [ [ x 1 , ... , x n ] ] be the ring of formal power series in variables x 1 , ... , x n with coefficients from K , let K [ [ x 1 , ... , x n ] ] → S be a K -algebra surjection and let E • • , • be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of S. Nicholas Switala [12] proved that this spectral sequence is independent of the surjection beginning with the E 2 page, and the groups E 2 p , q are all finite-dimensional over K. In this paper we extend this result to affine varieties. Namely, let Y be an affine variety over K , let X be a non-singular affine variety over K , let Y ⊂ X be an embedding over K and let E • • , • be the associated Hodge-de Rham spectral sequence for the computation of the de Rham homology of Y. Then this spectral sequence is independent of the embedding beginning with the E 2 page, and the groups E 2 p , q are all finite-dimensional over K. [ABSTRACT FROM AUTHOR]

The study of linear codes over local rings, particularly non-chain rings, imposes difficulties that differ from those encountered in codes over chain rings, and this stems from the fact that local non-chain rings are not principal ideal rings. In this paper, we present and successfully establish a new approach for linear codes of any finite length over local rings that are not necessarily chains. The main focus of this study is to produce generating characters, MacWilliams identities and generator matrices for codes over singleton local Frobenius rings of order 32. To do so, we first start by characterizing all singleton local rings of order 32 up to isomorphism. These rings happen to have strong connections to linear binary codes and Z 4 codes, which play a significant role in coding theory. [ABSTRACT FROM AUTHOR]