Injective Hull Research Articles

Morse subsets of injective spaces are strongly contracting

We show that a quasi-geodesic in an injective metric space is Morse if and only if it is strongly contracting. Since mapping class groups and, more generally, hierarchically hyperbolic groups act properly and coboundedly on injective metric spaces, we deduce various consequences relating, for example, to growth tightness and genericity of pseudo-Anosovs/Morse elements. Moreover, we show that injective metric spaces have the Morse local-to-global property and that a non-virtually cyclic group acting properly and coboundedly on an injective metric space is acylindrically hyperbolic if and only if contains a Morse ray. We also show that strongly contracting geodesics of a space stay strongly contracting in the injective hull of that space.

On modules in which every proper finitely generated submodule is a subset of a kernel of a nonzero endomorphism

We study an R-module M such that for every proper finitely generated submodule K of M, Hom R ( M / K , M ) ≠ 0 . Such modules are called co-finite-retractable (simply CFR). We consider when factor modules, direct sums and direct summands of CFR modules are CFR. Also, we investigate the injective hull of CFR modules. We explore certain properties of a ring R such that every (cyclic, free) module is CFR. Also, we show that R is a semisimple ring if and only if every R-module is regular if and only if every CFR module over R is regular.

On the graded injective hull of R/𝔭 in prime characteristic

On the graded injective hull of R/𝔭 in prime characteristic

Essential Extensions and Injective Hulls of Fuzzy Modules

In this paper, we introduce the notion of essential extensions of fuzzy modules. We use these concepts to introduce the notion of injective hulls of fuzzy modules. It is known that every [Formula: see text]-module has an injective hull, where [Formula: see text] is a ring. We show that these corresponding results do not hold for fuzzy [Formula: see text]-modules, i.e. there exists a fuzzy [Formula: see text]-module that does not have an injective hull. Sufficient conditions are given for a fuzzy [Formula: see text]-module to have an injective hull.

P-Rational Submodules

A submodule N is called rational in M if ( , E(M))=0, where E(M) is the an injective hull of M. Rational submodules have been studied and discussed by many authors such as H.H. Storrer, H. Khabazian, E. Ghashghaei, A. Hajikarimi, M.S. Abbas and M.S. Nayef. The main objective of this paper is to give a new class of submodules named P-rational submodules. This class is contained properly in the class of rational submodules. Several properties of this concept are introduced. The relationships between this class of submodules and some other related concepts are discussed such as essential and quasi-invertible submodules. Other characterizations of the P-rational submodule analogous to those which is known in the concept of the rational submodule are given.

On hypercyclic rings

Hypercyclic rings are those rings over which cyclic modules have cyclic injective hulls. Such rings were introduced by Caldwell in 1966 and studied by him and Osofsky in 1968. In this paper, we revisit Osofsky’s work on hypercyclic rings, highlighting some of her results and questions on the subject.

Injective Hulls are Completions of Ordered Algebras

Injective Hulls are Completions of Ordered Algebras

A remark on the category of graded 𝐹-modules

Let R = k [ x , y ] R=k[x,y] be a polynomial ring over a field k k of prime characteristic p p and let E E denote the injective hull of k k (which is isomorphic to H ( x , y ) 2 ( R ) H^2_{(x,y)}(R) ). We prove that E E is not an injective object in the category of graded F F -modules over R R . This answers in the negative a question raised by Lyubeznik-Singh-Walther [J. Eur. Math. Soc. (JEMS) 18 (2016), pp. 2545- 2578].

Critical and injective modules over skew polynomial rings

Let R be a commutative local k-algebra of Krull dimension one, where k is a field. Let α be a k-algebra automorphism of R, and define S to be the skew polynomial algebra R[θ;α]. We offer, under some additional assumptions on R, a criterion for S to have injective hulls of all simple S-modules locally Artinian - that is, for S to satisfy property (⋄). It is easy and well known that if α is of finite order, then S has this property, but in order to get the criterion when α has infinite order we found it necessary to classify all cyclic (Krull) critical S-modules in this case, a result which may be of independent interest. With the help of the above we show that Sˆ=k[[X]][θ,α] satisfies (⋄) for all k-algebra automorphisms α of k[[X]].

Quantaloidal Completions of Order-enriched Categories and Their Applications

By introducing the concept of quantaloidal completions for an order-enriched category, relationships between the category of quantaloids and the category of order-enriched categories are studied. It is proved that quantaloidal completions for an order-enriched category can be fully characterized as compatible quotients of the power-set completion. As applications, we show that a special type of injective hull of an order-enriched category is the MacNeille completion; the free quantaloid over an order-enriched category is the Down-set completion.

On generalized injective modules and almost injective modules

A module [Formula: see text] is called [Formula: see text]-almost-invariant for a module [Formula: see text] if, for any homomorphism [Formula: see text], either [Formula: see text], or there exist nonzero direct summands [Formula: see text] of [Formula: see text] and [Formula: see text] of [Formula: see text] such that [Formula: see text] is an isomorphism and [Formula: see text], where [Formula: see text] and [Formula: see text] are the injective hulls of [Formula: see text] and [Formula: see text], respectively. This is a generalization of an almost [Formula: see text]-injective module. In this paper, we give a new characterization of a generalized [Formula: see text]-injective module by homomorphisms between their injective hulls, and consider conditions for an [Formula: see text]-almost-invariant module to be almost [Formula: see text]-injective. Moreover, we study a relationship between generalized [Formula: see text]-injective, almost [Formula: see text]-injective and [Formula: see text]-almost-invariant modules.

Integral closure, basically full closure, and duals of nonresidual closure operations

We develop a duality for operations on nested pairs of modules that generalizes the duality between absolute interior operations and residual closure operations from [5], extending our previous results to the expanded context. We apply this duality in particular to integral and basically full closures and their respective cores to obtain integral and basically empty interiors and their respective hulls. We also dualize some of the known formulas for the core of an ideal to obtain formulas for the hull of a submodule of the injective hull of the residue field. The article concludes with illustrative examples in a numerical semigroup ring.

Reflectors to quantales

In this paper, we show that marked quantales have a reflection into quantales. To obtain the reflection we construct free quantales over marked quantales using appropriate lower sets. A marked quantale is a posemigroup in which certain admissible subsets are required to have joins, and multiplication distributes over these. Sometimes are the admissible subsets in question specified by means of a so-called selection function. A distinguishing feature of the study of marked quantales is that a small collection of axioms of an elementary nature allows one to do much that is traditional at the level of quantales. The axioms are sufficiently general to include as examples of marked quantales the classes of posemigroups, σ-quantales, prequantales and quantales. Furthermore, we discuss another reflection to quantales obtained by the injective hull of a posemigroup.

On modules in which every finitely generated submodule is a kernel of an endomorphism

We study an R-module M in which every finitely generated submodule of M is a kernel of an endomorphism of M. Such modules are called Co-epi-finite-retractable (CEFR). We also consider CEFR condition on the co-local modules, submodules and factors of a CEFR module and direct sum of CEFR modules. Among other results, we prove that the injective hull of a simple module over a commutative Noetherian ring is uniserial if and only if it is CEFR.We investigate modules over a principal ideal ring, and show that all finitely generated torsion modules over a principal ideal domain are CEFR. Also, we show that every module over a commutative Köthe ring is CEFR. We also observe that a ring R is left pseudo morphic if and only if it is CEFR as a left R-module and we obtain some new properties of left pseudo morphic rings.

On the topological structure of the (non-)finitely generated locus of Frobenius algebras emerging from Stanley–Reisner rings

Let U be the set of prime ideals P of the completion of a Stanley–Reisner ring S, such that the localization at P of the Frobenius algebra of the injective hull of the residue field of S is a finitely generated algebra. We give a partial answer to a conjecture made by M. Katzman about the openness of U. Specifically, we show that U has non-empty interior and we present some sufficient conditions for a principal open set D(f) to be contained in U, and for intersections of closed and principal opens to be contained in the complement of U.

On injective objects and existence of injective hulls in 𝑄-TOP/(𝑌, 𝜎)

In this paper, motivated by Cagliari and Mantovani, we have obtained a characterization of injective objects (with respect to the class of embeddings in the category 𝑄-TOP of 𝑄-topological spaces) in the comma category 𝑄-TOP/(𝑌,𝜎), when (𝑌,𝜎) is a stratified 𝑄-topological space, with the help of their 𝑇0-reflection. Further, we have proved that for any 𝑄-topological space (𝑌,𝜎), the existence of an injective hull of ((𝑋, 𝜏), 𝑓 ) in the comma category 𝑄-TOP/(𝑌, 𝜎) is equivalent to the existence of an injective hull of its 𝑇0-reflection ((𝑋 ̃,𝜏 ̃), 𝑓 ̃) in the comma category Q-TOP/(𝑌 ̃, 𝜎 ̃ ) (and in the comma category 𝑄-TOP0/(𝑌 ̃, 𝜎 ̃ ), where 𝑄-TOP0 denotes the category of 𝑇0-𝑄-topological spaces).

Injective Hulls of Various Graph Classes

A graph is Helly if its disks satisfy the Helly property, i.e., every family of pairwise intersecting disks in G has a common intersection. It is known that for every graph G, there exists a unique smallest Helly graph \(\mathcal {H}(G)\) into which G isometrically embeds; \(\mathcal {H}(G)\) is called the injective hull of G. Motivated by this, we investigate the structural properties of the injective hulls of various graph classes. We say that a class of graphs \(\mathcal {C}\) is closed under Hellification if \(G \in \mathcal {C}\) implies \(\mathcal {H}(G) \in \mathcal {C}\). We identify several graph classes that are closed under Hellification. We show that permutation graphs are not closed under Hellification, but chordal graphs, square-chordal graphs, and distance-hereditary graphs are. Graphs that have an efficiently computable injective hull are of particular interest. A linear-time algorithm to construct the injective hull of any distance-hereditary graph is provided and we show that the injective hull of several graphs from some other well-known classes of graphs are impossible to compute in subexponential time. In particular, there are split graphs, cocomparability graphs, and bipartite graphs G such that \(\mathcal {H}(G)\) contains \(\Omega (a^{n})\) vertices, where \(n=|V(G)|\) and \(a>1\).

Injective metrics on buildings and symmetric spaces

In this article, we show that the Goldman–Iwahori metric on the space of all norms on a fixed vector space satisfies the Helly property for balls. On the non-Archimedean side, we deduce that most classical Bruhat–Tits buildings may be endowed with a natural piecewise ℓ ∞ $\ell ^\infty$ metric which is injective. We also prove that most classical semisimple groups over non-Archimedean local fields act properly and cocompactly on Helly graphs. This gives another proof of biautomaticity for their uniform lattices. On the Archimedean side, we deduce that most classical symmetric spaces of non-compact type may be endowed with a natural invariant Finsler metric, restricting to an ℓ ∞ $\ell ^\infty$ metric on each flat, which is coarsely injective. We also prove that most classical semisimple groups over Archimedean local fields act properly and cocompactly on injective metric spaces. We identify the injective hull of the symmetric space of GL ( n , R ) $\operatorname{GL}(n,\mathbb {R})$ as the space of all norms on R n $\mathbb {R}^n$ . The only exception is the special linear group: if n = 3 $n=3$ or n ⩾ 5 $n \geqslant 5$ and K $\mathbb {K}$ is a local field, we show that SL ( n , K ) $\operatorname{SL}(n,\mathbb {K})$ does not act properly and coboundedly on an injective metric space.

Rings whose nonsingular right modules are $R$-projective

A right $R$-module $M$ is called $R$-projective provided that it is projective relative to the right $R$-module $R_{R}$. This paper deals with the rings whose all nonsingular right modules are $R$-projective. For a right nonsingular ring $R$, we prove that $R_{R}$ is of finite Goldie rank and all nonsingular right $R$-modules are $R$-projective if and only if $R$ is right finitely $\Sigma$-$CS$ and flat right $R$-modules are $R$-projective. Then, $R$-projectivity of the class of nonsingular injective right modules is also considered. Over right nonsingular rings of finite right Goldie rank, it is shown that $R$-projectivity of nonsingular injective right modules is equivalent to $R$-projectivity of the injective hull $E(R_{R})$. In this case, the injective hull $E(R_{R})$ has the decomposition $E(R_{R})=U_{R} \oplus V_{R}$, where $U$ is projective and $\operatorname{Hom}(V,R/I)=0$ for each right ideal $I$ of $R$. Finally, we focus on the right orthogonal class $\mathcal{N}^{\perp}$ of the class $\mathcal{N}$ of nonsingular right modules.

Locally finite representations over Noetherian Hopf algebras

We study finite dimensional representations over some Noetherian algebras over a field of characteristic zero. More precisely, we give necessary and sufficient conditions for the category of locally finite dimensional representations to be closed under taking injective hulls and extend results known for group rings and enveloping algebras to Ore extensions, Hopf crossed products, and affine Hopf algebras of low Gelfand-Kirillov dimension.