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In this paper, we study local rings from the perspective of reverse mathematics. We define local rings in a first-order way by using Π20 properties of invertible elements, where for a ring R possibly not commutative, R is left (resp. right) local if for any non-left (resp. non-right) invertible elements x, y ∈ R, x + y is not left (resp. right) invertible; R is local if for any non-invertible elements x, y ∈ R, x + y is not invertible. Firstly, we solve a question of Sato on characterizations of commutative local rings in his PhD thesis (Question 6.22 in Sato (2016)) and prove that the statement "a commutative ring is local if and only if it has at most one maximal ideal" is equivalent to ACA0 over RCA0. We also obtain a nice corollary in computable mathematics, i.e., there is a computable non-local ring with exactly two maximal ideals such that each of them Turing computes the Halting set K. Secondly, we study the equivalence among left local rings, right local rings, and local rings, showing that these three kinds of first-order local rings are equivalent over the weak basis theory RCA0. Finally, we extend the results of reverse mathematics on commutative local rings to noncommutative rings. [ABSTRACT FROM AUTHOR]

This paper explores the ideals and their structural properties in two generalizations of the partial transformation semigroup. Furthermore, principal, maximal, and minimal ideals within these semigroups are elucidated. [ABSTRACT FROM AUTHOR]

The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as "implies 0 = 1 ") we show, in constructive set theory with minimal logic, how for countable rings one can do without any kind of choice and without the usual decidability assumption that the ring is strongly discrete (membership in finitely generated ideals is decidable). By a functional recursive definition we obtain a maximal ideal in the sense that the quotient ring is a residue field (every noninvertible element is zero), and with strong discreteness even a geometric field (every element is either invertible or else zero). Krull's lemma for the related notion of prime ideal follows by passing to rings of fractions. By employing a construction variant of set-theoretic forcing due to Joyal and Tierney, we expand our treatment to arbitrary rings and establish a connection with dynamical algebra: We recover the dynamical approach to maximal ideals as a parametrized version of the celebrated double negation translation. This connection allows us to give formal a priori criteria elucidating the scope of the dynamical method. Along the way we do a case study for proofs in algebra with minimal logic, and generalize the construction to arbitrary inconsistency predicates. A partial Agda formalization is available at an accompanying repository.1 See https://github.com/iblech/constructive-maximal-ideals/. This text is a revised and extended version of the conference paper (In Revolutions and Revelations in Computability. 18th Conference on Computability in Europe (2022) Springer). The conference paper only briefly sketched the connection with dynamical algebra; did not compare this connection with other flavors of set-theoretic forcing; and sticked to the case of commutative algebra only, passing on the generalization to inconsistency predicates and well-orders. [ABSTRACT FROM AUTHOR]

In this paper, we study the structure of the skew polynomial ring R = ( p + u p) [ x ; ] and its quotient ring R n = R / 〈 x n − 1 〉 , where p is an odd prime number, u 2 = 0 , and (u) = − u. We give an explicit structure of the ideals in R and R n and propose an algorithm to characterize them. We identify the structure of prime, maximal, and primary ideals in these rings. In particular, we prove that this group ring is not Laskerian. [ABSTRACT FROM AUTHOR]

The ring of periodic distributions on R d with usual addition of distributions, and with convolution is considered. Via Fourier series expansions, this ring is isomorphic to the ring S ′ (Z d) of all maps f : Z d → C of at most polynomial growth (that is, there exist a real number M > 0 and an integer m ≥ 0 such that | f (n) | ≤ M (1 + | n 1 | + ⋯ + | n d |) m for all n = (n 1 , ⋯ , n d) ∈ Z d ), with pointwise operations. It is shown that finitely generated ideals in S ′ (Z d) are principal, and ideal membership is characterised analytically. Calling an ideal in S ′ (Z d) fixed if there is a common index n ∈ Z d where each member vanishes, the fixed maximal ideals are described, and it is shown that not all maximal ideals are fixed. It is shown that finitely generated (hence principal) prime ideals in S ′ (Z d) are fixed maximal ideals. The Krull dimension of S ′ (Z d) is proved to be infinite, while the weak Krull dimension is shown to be equal to 1. [ABSTRACT FROM AUTHOR]

AbstractLet R be a ring with unity. The upper ideal relation graph [Formula: see text] of the ring R is the simple undirected graph whose vertex set is the set of all non-unit elements of R and two distinct vertices x, y are adjacent if and only if there exists a non-unit element [Formula: see text] such that the ideals (x) and (y) contained in the ideal (z). In this article, we obtain the girth, minimum degree and the independence number of [Formula: see text]. We obtain a necessary and sufficient condition on R, in terms of the cardinality of their principal ideals, such that the graph [Formula: see text] is planar and outerplanar, respectively. For a non-local commutative ring [Formula: see text], where Ri is a local ring with maximal ideal [Formula: see text] and [Formula: see text], we prove that the graph [Formula: see text] is perfect if and only if [Formula: see text] and each [Formula: see text] is a principal ideal. We also discuss all the finite rings R such that the graph [Formula: see text] is Eulerian. Moreover, we obtain the metric dimension and strong metric dimension of [Formula: see text], when R is a reduced ring. Finally, we determine the vertex connectivity, automorphism group, Laplacian and the normalized Laplacian spectrum of [Formula: see text]. We classify all the values of n for which the graph [Formula: see text] is Hamiltonian.

The injective topic is well known for its influential relevance in module theory and scholars have worked hard to identify generalizations for it. One of these generalizations is M-Mininjective and mininjective, so we extend these notions to S-act theory as well, since act theory represents the generalization of module theory. If every S-homomorphism from a simple M-cyclic subact of M S into N S can be extended to M S , an S-act N S is called M-mininjective. An S-act M S is referred to as mininjective, if for each simple right ideal A of S and every S-homomorphism from A into M S can be extended to S-homomorphism from S into M S . We looked at the properties and characterizations of S-act where all subacts are M-cyclic and simple and all subacts are merely simple. These topics are shown using examples. With the provided concepts, we were able to accomplish improved results, obtaining novel characterizations of mininjective acts in terms of duality conditions. Additionally, the conditions under which subacts inherit the mininjective and M-mininjective properties are studied. The connection between the act of maximal right ideals of S and the act of minimal subact of T M is explicated. Finally, the conditions under which the classes of M-mininjective acts and the classes of mininjective S-acts will coincide are defined. Our work's conclusions have been explained. [ABSTRACT FROM AUTHOR]

In this paper, we study the maximal ideals in a commutative ring of bicomplex numbers and then we describe the maximal ideals in a bicomplex algebra. We found that the kernel of a nonzero multiplicative B C -linear functional in a commutative bicomplex Banach algebra need not be a maximal ideal. Finally, we introduce the notion of bicomplex division algebra and generalize the Gelfand–Mazur theorem for the bicomplex division Banach algebra. [ABSTRACT FROM AUTHOR]

Significant stochastic orders can be characterized by other stochastic orders when probabilities are transformed by means of measurable mappings which fulfils certain conditions. The main aim of this manuscript is to provide a unified approach of those orders. Our study shows how to obtain important results of such orders by means of conditions on the underlying stochastic orders. Those results are mainly in relation to maximal generators, to transition kernels between ordered probabilities, and to probabilistic operators of kernels. [ABSTRACT FROM AUTHOR]

In this paper, we will now develop the content of quasi-commutative near_subtraction semi groups, Semi prime ideals, Proper prime ideals, globally idempotent principal ideals. Relation between proper prime ideals and semi prime ideals. [ABSTRACT FROM AUTHOR]

In an Almost Distributive Fuzzy Lattice (ADFL), the concept of S-ideals is introduced, and some significant features of these ideals are derived. S-ideals are used to categorise ADFLs. Prime ideals were also confirmed in ADFL. For the class of all S-ideals to become a DFL to the lattice of all ideals, a set of analogous requirements is stated, leading to the characterization of an ADFL. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. First, to provide a characterization of clopen topologies. We show that a topology is clopen if and only if it is symmetric and Alexandroff. Second, to investigate when some special unitary subrings of a power set define a (clopen) topology. [ABSTRACT FROM AUTHOR]

This paper presents a PVS development of relevant results of the theory of rings. The PVS theory includes complete proofs of the three classical isomorphism theorems for rings, and characterizations of principal, prime and maximal ideals. Algebraic concepts and properties are specified and formalized as generally as possible allowing in this manner their application to other algebraic structures. The development provides the required elements to formalize important algebraic theorems. In particular, the paper presents the formalization of the general algebraic-theoretical version of the Chinese remainder theorem (CRT) for the theory of rings, as given in abstract algebra textbooks, proved as a consequence of the first isomorphism theorem. Also, the PVS theory includes a formalization of the number-theoretical version of CRT for the structure of integers, which is the version of CRT found in formalizations. CRT for integers is obtained as a consequence of the general version of CRT for the theory of rings. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of a fuzzy weakly irreducible ideal of a commutative ring R with identity. This concept is a generalization of the concept of a fuzzy strongly irreducible ideal. Also, relationships between fuzzy maximal, fuzzy quasi primary and fuzzy weakly irreducible ideals in a ring are proved. [ABSTRACT FROM AUTHOR]

In this paper, we discuss several properties of ideals, filters, annihilators and maximisors in a general Almost Distributive Lattices (ADLs) and in particular, Almost Boolean Algebras (ABAs). Also, we characterize extreme ideals and filters. Further, several equivalent conditions are obtained in terms of ideals and filters for an ADL to become an ABA. [ABSTRACT FROM AUTHOR]

The concept of radicals of ideals is introduced in BE-algebras and certain properties of these radicals are derived in terms of direct products and homomorphisms. The notion of semi-maximal ideals is introduced in BE-algebras through the radical of ideals and their significant properties are investigated. Some equivalent conditions are derived for every semi-maximal ideal to become a maximal ideal. [ABSTRACT FROM AUTHOR]

Let be an ultrametric complete valued field and be an ultrametric space. We examine some Banach algebras of bounded continuous functions from to with the use of ultrafilters, particularly the relation of stickness. We recall and deepen results obtained in a previous paper by N. Maïnetti and the third author concerning the whole algebra of all bounded continuous functions from to . Every maximal ideal of finite codimension of is of codimension and we show that this property also holds for every algebra , provided is perfect. If admits the uniform norm on as its spectral norm, then every maximal ideal is the kernel of only one multiplicative semi-norm, the Shilov boundary is equal to the whole multiplicative spectrum and the Banaschewski compactification of is homeomorphic to the multiplicative spectrum of . [ABSTRACT FROM AUTHOR]

The core of the paper is to investigate the class of generalization of quasi injective S-acts where we generalized the notion of quasi-injective act to several types. Minimal quasi injective S-act is introduced and studied. More precisely, we study properties and characterizations of S-acts in which all subacts are simple. We highlight a relationship of the concept of minimal quasi injective S-acts with min-annihilator acts, min-symmetric acts. We give a characterization of minimal quasi injective acts in terMs of duality. In addition, we investigate conditions under which subacts inherit the minimal quasi injective property. We prove that for strongly Kasch S-act Ms, if Ms is a minimal quasi injective S-act, then there is a bijection between the class of minimal subacts of TM and the class of maximal right ideals of its endomorphism monoid S. We give some characterizations and properties of the structure of endomorphism monoid of minimal quasi injective acts and a minimal quasi injective monoid and then mention the relationship between them. Finally, we study the relationship between the act of all maximal right ideals of S and the act of minimal subacts of TM. [ABSTRACT FROM AUTHOR]

In this paper, we define and characterize the notions of (implicative, maximal, prime) ideals in hoops. Then we investigate the relation between them and prove that every maximal implicative ideal of a V-hoop with double negation property is a prime one. Also, we define a congruence relation on hoops by ideals and study the quotient that is made by it. This notion helps us to show that an ideal is maximal if and only if the quotient hoop is a simple MV-algebra. Also, we investigate the relationship between ideals and filters by exploiting the set of complements. [ABSTRACT FROM AUTHOR]

Given an ideal I on the positive integers, a real sequence (x n) is said to be I -statistically convergent to ℓ provided that n ∈ N : 1 n | { k ≤ n : x k ∉ U } | ≥ ε ∈ I for all neighborhoods U of ℓ and all ε > 0. First, we show that I -statistical convergence coincides with J -convergence, for some unique ideal J = J (I). In addition, J is Borel [analytic, coanalytic, respectively] whenever I is Borel [analytic, coanalytic, resp.]. Then we prove, among others, that if I is the summable ideal { A ⊆ N : ∑ a ∈ A 1 ∕ a < ∞ } or the density zero ideal { A ⊆ N : lim n → ∞ 1 n | A ∩ [ 1 , n ] | = 0 } then I -statistical convergence coincides with statistical convergence. This can be seen as a Tauberian theorem which extends a classical theorem of Fridy. Lastly, we show that this is never the case if I is maximal. [ABSTRACT FROM AUTHOR]

The aim of this paper is to introduce and study a new kind of graphs associated to an ideal of a commutative ring. Let R be a commutative ring with identity, and I(R) be the set of all non-trivial ideals of R with S∈I(R). The sum ideal graph associated to S, denoted by Ψ(R, S), is the undirected graph with vertex set {A∈I(R): S⊂A+B, for some B∈I(R)} where two ideal vertices A and B are adjacent if and only if A B and S⊂A+B. In this paper we establish some of characterizations and results of this kind of graph with providing some examples. [ABSTRACT FROM AUTHOR]

Let IE be a complete ultrametric space, let IK be a perfect complete ultrametric field and let A be a Banach IK-algebra which is either a full IK-subalgebra of the algebra of continuous functions from IE to IK owning all characteristic functions of clopens of IE, or a full IK-subalgebra of the algebra of uniformly continuous functions from IE to IK owning all characteristic functions of uniformly open subsets of IE. We prove that all maximal ideals of finite codimension of A are of codimension 1. [ABSTRACT FROM AUTHOR]

Abstract Let D be a Dedekind domain with infinitely many maximal ideals, all of finite index, and K its quotient field. Let Int (D) = { f ∈ K [ x ] | f (D) ⊆ D } be the ring of integer-valued polynomials on D. Given any finite multiset { k 1 , ... , k n } of integers greater than 1, we construct a polynomial in Int (D) which has exactly n essentially different factorizations into irreducibles in Int (D) , the lengths of these factorizations being k 1 , ..., k n. We also show that there is no transfer homomorphism from the multiplicative monoid of Int (D) to a block monoid. [ABSTRACT FROM AUTHOR]

Abstract There are very few results about maximal d.r.e. degrees as the construction is very hard to work with other requirements. In this paper we show that there exists an isolated maximal d.r.e. degree. In fact, we introduce a closely related notion called (m , n) -cupping degree and show that there exists an isolated (2 , ω) -cupping degree, and there exists a proper (2 , 1) -cupping degree. It helps understanding various degree structures in the Ershov Hierarchy. [ABSTRACT FROM AUTHOR]

We prove a uniqueness theorem for irreducible non-critical Gelfand–Tsetlin modules. The uniqueness result leads to a complete classification of the irreducible Gelfand–Tsetlin modules with 1-singularity. An explicit construction of such modules was given in Futorny et al. [ 7 ]. In particular, we show that the modules constructed in Futorny et al. [ 7 ] exhaust all irreducible Gelfand–Tsetlin modules with 1-singularity. To prove the result, we introduce a new category of modules (called Drinfeld category) related to the Drinfeld generators of the Yangian Y(gln) and define a functor from the category of non-critical Gelfand–Tsetlin modules to the Drinfeld category. [ABSTRACT FROM AUTHOR]

We give a characterization of conformal classes realizing a compact manifold's Yamabe invariant. This characterization is the analogue of an observation of Nadirashvili for metrics realizing the maximal first eigenvalue, and of Fraser and Schoen for metrics realizing the maximal first Steklov eigenvalue. [ABSTRACT FROM AUTHOR]

Characterization of quasi-complemented ADLs in terms of its α-ideals are studied. Some sufficient conditions of weakly disjunctive ADL to become quasi-comp-lemented ADLs are derived. [ABSTRACT FROM AUTHOR]

This text by pure algebraic reasons outlines why the spectrum of maximal ideals SpecℂA of a countable-dimensional differential ℂ-algebra A of transcendence degree 1 without zero divisors is locally analytic, which means that for any ℂ-homomorphism ψM : A → ℂ(M ∈ SpecℂA) and any a ∈ A the Taylor series ψ∼Ma=def∑m=0∞ψMamzmm! has nonzero radius of convergence depending on the element a ∈ A. [ABSTRACT FROM AUTHOR]

Let E be an arbitrary directed graph and let L be the Leavitt path algebra of the graph E over a field K. The necessary and sufficient conditions are given to assure the existence of a maximal ideal in L and also the necessary and sufficient conditions on the graph that assure that every ideal is contained in a maximal ideal are given. It is shown that if a maximal ideal M of L is nongraded, then the largest graded ideal in M, namely gr(M), is also maximal among the graded ideals of L. Moreover, if L has a unique maximal ideal M, then M must be a graded ideal. The necessary and sufficient conditions on the graph for which every maximal ideal is graded are discussed. [ABSTRACT FROM AUTHOR]

For the elements of a numerical semigroup which are larger than the Frobenius number, we introduce the definition of seed by broadening the notion of generator. This new concept allows us to explore the semigroup tree in an alternative efficient way, since the seeds of each descendant can be easily obtained from the seeds of its parent. The paper is devoted to presenting the results which are related to this approach, leading to a new algorithm for computing and counting the semigroups of a given genus. [ABSTRACT FROM AUTHOR]

In this paper we develop a theory of monomial preorders, which differ from the classical notion of monomial orders in that they allow ties between monomials. Since for monomial preorders, the leading ideal is less degenerate than for monomial orders, our results can be used to study problems where monomial orders fail to give a solution. Some of our results are new even in the classical case of monomial orders and in the special case in which the leading ideal defines the tangent cone. [ABSTRACT FROM AUTHOR]

Assuming Generalized Riemann's Hypothesis, Bach proved that the class group clκ of a number field K may be generated using prime ideals whose norm is bounded by 12 log2 Δκ, and by (4 + o(1)) log2 Δκ asymptotically, where Δκ is the absolute value of the discriminant of K. Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates clκ and which performs better than Bach's bound in computations, but which is asymptotically worse. In this paper we show that clκ is generated by prime ideals whose norm is bounded by the minimum of 4.01 log2 Δκ, 4(1 + (2πeγ)-N)K)2 log2 Δκ and 4(logΔκ +log logΔκ -(γ +log 2π)Nκ +1+(Nκ +1) log(7 logΔκ) logΔκ)2. Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedman's algorithms, confirming that it has size ... (logΔκ log logΔκ)2. In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be smaller than log2 Δκ except for 7 out of the 31292 fields we tested. [ABSTRACT FROM AUTHOR]

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An almost real maximal ideal M of C ( X ) is a maximal ideal that is either fixed or Z [ M ] contains a free z -filter which is closed under countable intersection. Using these maximal ideals, we first construct a space λX which is weakly Lindelöf containing X as a dense subspace on which every f ∈ C ( X ) with Lindelöf cozero-set can be extended (when this is the case, we say that X is C L -embedded). Next, using this space, we present the largest Lindelöf subspace Λ X of βX in which X is C L -embedded. If X is locally Lindelöf, λX coincides with Λ X and it turns out in this case that Λ X ( = λ X ) is the smallest Lindelöf subspace of βX with compact remainder. Using the structure of Λ X , we also give the smallest realcompact subspace ϒ X of βX with compact remainder. Finally the relations between the spaces υX , λX , ϒ X , Λ X and βX are investigated and we apply the structures of these spaces to characterize some intersections of free maximal ideals of C ( X ) . [ABSTRACT FROM AUTHOR]

Let J65 be the Jacobian of the Shimura curve attached to the indefinite quaternion algebra over Q of discriminant 65. We study the isogenies J0(65) → J65 defined over Q, whose existence was proved by Ribet. We prove that there is an isogeny whose kernel is supported on the Eisenstein maximal ideals of the Hecke algebra acting on J0(65), and, moreover, the odd part of the kernel is generated by a cuspidal divisor of order 7, as is predicted by a conjecture of Ogg. [ABSTRACT FROM AUTHOR]

We investigate three kinds of generalized kernels of an ordered semigroup. Those that are the intersection of all prime ideals, the intersection of all maximal ideals, and the intersection of all completely prime ideals. We obtain a structure of an ordered semigroup for which the intersection of all prime ideals coincides with the intersection of all maximal ideals. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate the prime, primary, and maximal ideals of semirings. The localization and primary decomposition of ideals in semirings are also studied. [ABSTRACT FROM AUTHOR]

One knows that the concept of minimality and maximality of left ideals and right ideals play an important role in semigroups. In this paper, we extend this concept to consider in n-ary semigroups. A number of results concerning relationships between minimality and maximality of n-ideals of n-ary semigroups and n-simple (0-n-simple) n-ary semigroups as well as some characterizations of minimality and maximality of n-ideals of n-ary semigroups are given. [ABSTRACT FROM AUTHOR]

A depiction of a nonnoetherian integral domain R is a special coordinate ring that provides a framework for describing the geometry of R. We show that if R is noetherian in codimension 1, then R has a unique maximal depiction T. In this case, the geometric dimensions of the points of Spec R may be computed directly from T. If in addition R has a normal depiction S, then S is the unique maximal depiction of R. [ABSTRACT FROM AUTHOR]

We describe the endomorphism rings in an additive category whose objects are right R-modules M with a fixed chain of submodules and the behavior of these objects as far as their direct sums are concerned. [ABSTRACT FROM AUTHOR]

We use a landmark result in the theory of Riesz spaces – Freudenthal’s 1936 spectral theorem – to canonically represent any Archimedean lattice-ordered group G with a strong unit as a (non-separating) lattice-group of real-valued continuous functions on an appropriate G-indexed zero-dimensional compactification w G ⁢ Z G {w_{G}Z_{G}} of its space Z G {Z_{G}} of minimal prime ideals.The two further ingredients needed to establish this representation are the Yosida representation of G on its space X G {X_{G}} of maximal ideals, and the well-known continuous surjection of Z G {Z_{G}} onto X G {X_{G}}.We then establish our main result by showing that the inclusion-minimal extension of this representation of G that separates the points of Z G {Z_{G}} – namely, the sublattice subgroup of C ⁡ ( Z G ) {\operatorname{C}(Z_{G})} generated by the image of G along with all characteristic functions of clopen (closed and open) subsets of Z G {Z_{G}} which are determined by elements of G – is precisely the classical projectable hull of G.Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit. [ABSTRACT FROM AUTHOR]

For subsets of ℝ+=[0,∞) we introduce a notion of coherently porous sets as the sets for which the upper limit in the definition of porosity at a point is attained along the same sequence. We prove that the union of two strongly porous at 00 sets is strongly porous if and only if these sets are coherently porous. This result leads to a characteristic property of the intersection of all maximal ideals contained in the family of strongly porous at 00 subsets of ℝ+. It is also shown that the union of a set A⊆R+A⊆ℝ+ with arbitrary strongly porous at 00 set is porous at 00 if and only if AA is lower porous at 00. [ABSTRACT FROM AUTHOR]

A prime ideal is said to be strongly prime if whenever contains an intersection of ideals, contains one of the ideals in the intersection. A commutative ring with this property for every prime ideal is called strongly zero-dimensional. Some equivalent conditions are given and it is proved that a zero-dimensional ring is strongly zero-dimensional if and only if the ring is quasi-semi-local. A ring is called strongly -regular if in each ideal , there is an element such that for all . Connections between the concepts strongly zero-dimensional and strongly -regular are considered. [ABSTRACT FROM AUTHOR]

We study a large family of products of Borel fixed ideals of maximal minors. We compute their initial ideals and primary decompositions, and show that they have linear free resolutions. The main tools are an extension of straightening law and a very uniform primary decomposition formula. We study also the homological properties of associated multi-Rees algebra which are shown to be Cohen–Macaulay, Koszul and defined by a Gröbner basis of quadrics. [ABSTRACT FROM AUTHOR]

Let be a Noetherian local ring with maximal ideal and residue field . The linearity defect of a finitely generated -module , which is denoted , is a numerical measure of how far is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether implies , in the case when . [ABSTRACT FROM AUTHOR]

Let S be a polynomial ring over a field K of characteristic zero and let M ⊂ S be an ideal given as an intersection of powers of incomparable monomial prime ideals (e.g., the case where M is a squarefree monomial ideal). In this paper we provide a very effective, sufficient condition for a monomial prime ideal P ⊂ S containing M be such that the localisation MP has non-maximal analytic spread. Our technique describes, in fact, a concrete obstruction for P to be an asymptotic prime divisor of M with respect to the integral closure filtration, allowing us to employ a theorem of McAdam as a bridge to analytic spread. As an application, we derive – with the aid of results of Brodmann and Eisenbud-Huneke – a situation where the asymptotic and conormal asymptotic depths cannot vanish locally at such primes. [ABSTRACT FROM PUBLISHER]

A topological space is called submaximal if each of its dense subsets is open and is called nodec if each of its nowhere dense subsets is closed. Here, we study a variety of spaces some of which have already been studied in C(X). Among them are, most importantly, quasi P-spaces and pointwise quasi P-spaces. We obtain some new useful topological characterizations of quasi P-spaces and pointwise quasi P-spaces. Consequently, we obtain a close relation between these latter spaces and submaximal and nodec spaces. [ABSTRACT FROM AUTHOR]