Essential Extensions and Injective Hulls of Fuzzy Modules

The definition of injectivity, and the proof that every module has an injective extension which is a subextension of every other injective extension, are due to R. Baer [B]. An independent proof using the notion of essential extension was given by Eckmann-Schopf [ES]. Both proofs require the p reliminary construction of some injective overmodule. In [F] I showed how the latter proof could be freed from this requirement by exhibiting a set F in which every essential extension could be embedded. Subsequently J. M. Maranda pointed out that F has minimal cardinality. It follows that F is equipotent with the injective hull. Below Icon struct the injective hull by equipping Fit self with a module strucure.

The paper consists of two parts. In the first part the concepts of injective objects, essential extensions, and injective hulls are illustrated by examples of known characterizations of these concepts in various familiar settings. In the second part the above notions are considered in quasicategories whose objects are concrete categories. It turns out that several types of ‘completions’ of concrete categories can be characterized as injective hulls.

The homological theory of rings and modules is an important branch of algebra. It provided answers to numerous questions of the theory of rings. Along with the homological theory, another theory started to develop, also under significant influence of the theory of rings, which is the homological theory of universal algebras, and, in particular, of semigroups and acts over them. This theory analyses such notions as injective and projective acts over semigroups, injective hulls and projective covers. As in the case of rings and modules, the injective hull exists for every act, while the projective cover sometimes does not. In 1967 P. Berthiaume proved the existence of injective hulls of an arbitrary act over a semigroup (without the assumption of the presence of an identity in the semigroup). J. Isbell studied monoids (i.e. semigroups with an identity) over which every act has a projective cover. L. A. Skornyakov developed a homological theory of monoids. Many results of that theory were mentioned in the known monograph by M. Kilp, U. Knauer, A. V. Mikhalev. For semigroups of a relatively simple structure the results of the homological theory can be significantly refined. For example, in 2012 G. Moghaddasi described injective acts and built injective hulls of acts over a left zero semigroup assuming the separability of the act. I. B. Kozhukhov and A. P. Haliullina described injective and projective acts over groups and right zero semigroups, built injective hulls and projective covers of acts over such semigroups. For acts over a left zero semigroup the condition of separability of acts was removed. An important class of semigroups containing groups, left and right zero semigroups, rectangular bands is the class of completely simple semigroups, as well as the broader class of completely 0-simple semigroups. In 2000 A. Yu. Avdeyev and I. B. Kozhukhov described all acts over completely simple semigroups and acts with zero over completely 0-simple semigroups. It triggered further reasearch of acts over such semigroups. I. B. Kozhuhov and A. O. Petrikov described injective and projective acts over completely simple semigroups, thereby generalising the results of I. B. Kozhuhov and A. R. Khaliullina, and also the work of G. Mogaddasi. They built injective hulls and projective covers of acts over such semigroups. In this paper the above-mentioned results concerning acts over completely simple semigroups were generalized to acts with zero over completely 0-simple semigroups. In particular, the necessary and sufficient conditions of injectivity and projectivity of an act with zero over an arbitrary completely 0-simple semigroup were found, injective hulls and projective covers of arbitrary acts with zero over such semigroups were built. It was established that a projective act over an arbitrary completely 0-simple semigroup is exactly a 0-coproduct of a free act and acts isomorphic to a 0-minimal right ideal of the semigroup (considered as a right act).

A module is quasi-injective if and only if it is invariant under endomorphisms of its injective hull. In this paper, we study the class of modules which are invariant under all clean endomorphisms of their injective hulls and show that this class of modules coincide with the class of quasi-injective modules. Some facts and results of this class of modules are obtained. We also establish some relations of clean-invariant modules with automorphism-invariant modules, idempotent-invariant modules, pseudo-continuous modules, and Utumi modules. Apart from these, we have given several sufficient conditions under which automorphism-invariant modules can be clean-invariant.KeywordsAutomorphism-invariant modulesIdempotent-invariant modulesClean-invariant modules

It is well known that the injective hull of a lattice-ordered group ( l l -group) M M can be given a lattice order in a unique way so that it becomes an l l -group extension of M M . This is not the case for an arbitrary f f -module over a partially ordered ring (po-ring). The fact that it is the case for any l l -group is used extensively to get deep theorems in the theory of l l -groups. For instance, it is used in the proof of the Hahn-embedding theorem and in the characterization of ℵ a {\aleph _a} -injective l l -groups. In this paper we give a necessary and sufficient condition on the injective hull of a torsion-free f f -module M M (over a directed essentially positive po-ring) for it to be made into an f f -module extension of M M (in a unique way). An f f -module is called an i − f i - f -module if its injective hull can be made into an f f -module extension. The class of torsion-free i − f i - f -modules is closed under the formation of products, sums, and Hahn products of strict f f -modules. Also, an l l -submodule and a torsion-free homomorphic image of a torsion-free i − f i - f -module are i − f i - f -modules. Let R R be an f f -ring with zero right singular ideal whose Boolean algebra of polars is atomic. We show that R R is a q f qf -ring (i.e., R R {R_R} is an i − f i - f -module) if and only if each torsion-free R − f R - f -module is an i − f i - f -module. There are no injectives in the category of torsion-free R − f R - f -modules, but there are ℵ a {\aleph _a} -injectives. These may be characterized as the f f -modules that are injective R R -modules and ℵ a {\aleph _a} -injective l l -groups. In addition, each torsion-free f f -module over R R can be embedded in a Hahn product of l l -simple Q ( R ) − f Q(R) - f -modules. We note, too, that a totally ordered domain has an i − f i - f -module if and only if it is a right Ore domain.

The Zariski topology and essential extensions of semilattices

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Let R M _RM be a left R R -module such that Hom R ( M , U ) ≠ 0 {\operatorname {Hom} _R}(M,U) \ne 0 for any nonzero submodule U U of M M , let E ( M ) E(M) denote the injective hull of M M , and let B B (resp. A A ) denote the ring of R R -endomorphisms of M M (resp. E ( M ) E(M) ). It is known that if M M is nonsingular then B B is left nonsingular and A A is the maximal left quotient ring of B B . We give here necessary and sufficient conditions on M M for B B to be right nonsingular and for A A to be the maximal right quotient ring of B B .

In feedback stabilization studies, a control scheme is designed so that the ‘state’ of the system decays with a designated rate as t → ∞. Thus, any linear functional of the state also decays with at least the same decay rate. We study a class of linear parabolic systems, and show the existence of a feedback control scheme with a specific property such that some non-trivial linear functionals decay faster than the state. In constructing the control scheme, we also solve the new problem of an arbitrary allocation of the eigenvalues of some coefficient matrix which is subject to constraint, and give the necessary and sufficient condition in terms of controllability and observabililty assumptions. This is an essential extension of the celebrated 1967 theory of pole allocation by W. M. Wonham.

In the category of T1 -spaces and continuous maps, every space X has a unique maximal essential extension. X has an injective hull iff card X≤1.

For a ring R, there are classical facts that R is right Noetherian if and only if every direct sum of injective right R-modules is injective, and R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of injective right R-modules. In this paper, we prove that R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of either injective right R-modules or projective right R-modules.

Let be a nonempty subset of the set of submodules of a module M . Then M is called a -extending module if for each X in there exists a direct summand D of M such that X is essential in D. In general, it is known that an essential extension of a -extending module is not -extending. In this paper, our goal is to show how to construct essential extensions of a module which are extending by using a set of representatives of a certain equivalence relation on the set of all idempotent endomorphisms of the injective hull of the module. Also we characterize when the rational hull of a module is -extending in terms of such a set of representatives. Moreover we show that several well known types of -extending conditions (e.g., extending and FI-extending) transfer from module to its rational hull.