Hilbert \(C^{*}\)-modules in which all relatively strictly closed submodules are complemented

We give the characterization and description of all full Hilbert modules and associated algebras having the property that each relatively strictly closed submodule is orthogonally complemented. A strict topology is determined by an essential closed two-sided ideal in the associated algebra and a related ideal submodule. It is shown that these are some modules over hereditary algebras containing the essential ideal isomorphic to the algebra of (not necessarily all) compact operators on a Hilbert space. The characterization and description of that broader class of Hilbert modules and their associated algebras is given. As auxiliary results we give properties of strict and relatively strict submodule closures, the characterization of orthogonal closedness and orthogonal complementing property for single submodules, relation of relative strict topology and projections, properties of outer direct sums with respect to the ideals in \(\ell_\infty\) and isomorphisms of Hilbert modules, and we prove some properties of hereditary algebras and associated hereditary modules with respect to the multiplier algebras, multiplier Hilbert modules, corona algebras and corona modules.

Summand intersection property of modules via z-closed submodules

In this paper, we define a module [Formula: see text] to be [Formula: see text] if and only if intersection of each pair of [Formula: see text]-closed direct summands is also a direct summand of [Formula: see text]. We investigate structural properties of [Formula: see text]-modules and locate the implications between the other module properties which are essentially based on direct summands. We deal with decomposition theory as well as direct summands of [Formula: see text]-modules. We apply our results to matrix rings. To this end, it is obtained that the [Formula: see text] property is not Morita invariant.

Projectivity relative to closed (neat) submodules

An [Formula: see text]-module [Formula: see text] is called closed (neat) projective if, for every closed (neat) submodule [Formula: see text] of every [Formula: see text]-module [Formula: see text], every homomorphism from [Formula: see text] to [Formula: see text] lifts to [Formula: see text]. In this paper, we study closed (neat) projective modules. In particular, the structure of a ring over which every finitely generated (cyclic, injective) right [Formula: see text]-module is closed (neat) projective is studied. Furthermore, the relationship among the proper classes which are induced by closed submodules, neat submodules, pure submodules and [Formula: see text]-pure submodules are investigated.

Some Properties of the Essential Fuzzy and Closed Fuzzy Submodules

In this paper, we introduce and study the essential and closed fuzzy submodules of a fuzzy module X as a generalization of the notions of essential and closed submodules. We prove many basic properties of both concepts.

Proper classes generated by τ- closed submodules

The main object of this paper is to study relative homological aspects as well as further properties of τ -closed submodules. A submodule N of a module M is said to be τ -closed (or τ -pure) provided that M/N is τ -torsion-free, where τ stands for an idempotent radical. Whereas the well-known proper class 𝒞losed (𝒫ure) of closed (pure) short exact sequences, the class τ −𝒞losed of τ -closed short exact sequences need not be a proper class. We describe the smallest proper class 〈τ − 𝒞losed〉 containing τ − 𝒞losed, through τ -closed submodules. We show that the smallest proper class 〈τ − 𝒞losed〉 is the proper classes projectively generated by the class of τ -torsion modules and coprojectively generated by the class of τ -torsion-free modules. Also, we consider the relations between the proper class 〈τ − 𝒞losed〉 and some of well-known proper classes, such as 𝒞losed, 𝒫ure.

W-Closed Submodule and Related Concepts

Let R be a commutative ring with identity, and M be a left untial module. In this paper we introduce and study the concept w-closed submodules, that is stronger form of the concept of closed submodules, where asubmodule K of a module M is called w-closed in M, "if it has no proper weak essential extension in M", that is if there exists a submodule L of M with K is weak essential submodule of L then K=L. Some basic properties, examples of w-closed submodules are investigated, and some relationships between w-closed submodules and other related modules are studied. Furthermore, modules with chain condition on w-closed submodules are studied.