Direct Sums of Prime Subsemimodules

Let be a commutative semiring. A semimodule over a semiring is a fully prime semimodule if each proper subsemimodule of is prime. This research aims to investigate the relationship between a direct sum of prime subsemimodules and , , and a fully prime semimodule.

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.

New Results for Arithmetic –Geometric Mean Inequalities and Singular Values in Matrices

Matrix theory is very popular in different kind of sciences such as engineering, architecture, physics, chemistry, computer science, IT, so on as well as mathematics many theoretical results dealing with the structure of the matrices even this topic seems easy to work. That is why many scientists still consider some open problem in matrix theory. In this paper, generalizations of the arithmetic-geometric mean inequality is presented for singular values related to block matrices. Singular values are also given for sums, products and direct sums of the matrices.

The Covering Radius and a Discrete Surface Area for Non-Hollow Simplices

We explore upper bounds on the covering radius of non-hollow lattice polytopes. In particular, we conjecture a general upper bound of d/2 in dimension d, achieved by the “standard terminal simplices” and direct sums of them. We prove this conjecture up to dimension three and show it to be equivalent to the conjecture of González-Merino and Schymura (Discrete Comput. Geom. 58(3), 663–685 (2017)) that the d-th covering minimum of the standard terminal n-simplex equals d/2, for every $$n\ge d$$ n ≥ d . We also show that these two conjectures would follow from a discrete analog for lattice simplices of Hadwiger’s formula bounding the covering radius of a convex body in terms of the ratio of surface area versus volume. To this end, we introduce a new notion of discrete surface area of non-hollow simplices. We prove our discrete analog in dimension two and give strong evidence for its validity in arbitrary dimension.

Relative Essential Ideals in N-groups

Let $G$ be an $N$-group where $N$ is a (right) nearring. We introduce the concept of relative essential ideal (or $N$-subgroup) as a generalization of the concept of essential submodule of a module over a ring or a nearring. We provide suitable examples to distinguish the notions relative essential and essential ideals. We prove the important properties and obtain equivalent conditions for the relative essential ideals (or $N$-subgroups) involving the quotient. Further, we derive results on direct sums, complement ideals of $N$-groups and obtain their properties under homomorphism.

On strongly coseparable modules

A module M is called $$\mathfrak {s}$$ s -coseparable if for every nonzero submodule U of M such that M/U is finitely generated, there exists a nonzero direct summand V of M such that $$V \subseteq U$$ V ⊆ U and M/V is finitely generated. It is shown that every non-finitely generated free module is $$\mathfrak {s}$$ s -coseparable but a finitely generated free module is not, in general, $$\mathfrak {s}$$ s -coseparable. We prove that the class of $$\mathfrak {s}$$ s -coseparable modules over a right noetherian ring is closed under finite direct sums. We show that the class of commutative rings R for which every cyclic R-module is $$\mathfrak {s}$$ s -coseparable is exactly that of von Neumann regular rings. Some examples of modules M for which every direct summand of M is $$\mathfrak {s}$$ s -coseparable are provided.

Abelian p-groups with minimal full inertia

The class of abelian p-groups with minimal full inertia, that is, satisfying the property that fully inert subgroups are commensurable with fully invariant subgroups is investigated, as well as the class of groups not satisfying this property; it is known that both the class of direct sums of cyclic groups and that of torsion-complete groups are of the first type. It is proved that groups with “small" endomorphism ring do not satisfy the property and concrete examples of them are provided via Corner’s realization theorems. Closure properties with respect to direct sums of the two classes of groups are also studied. A topological condition of the socle and a structural condition of the Jacobson radical of the endomorphism ring of a p-group G, both of which are satisfied by direct sums of cyclic groups and by torsion-complete groups, are shown to be independent of the property of having minimal full inertia. The new examples of fully inert subgroups, which are proved not to be commensurable with fully invariant subgroups, are shown not to be uniformly fully inert.

Strongly Gorenstein categories

We introduce and study strongly Gorenstein subcategory [Formula: see text], relative to an additive full subcategory [Formula: see text] of an abelian category [Formula: see text]. When [Formula: see text] is self-orthogonal, we give some sufficient conditions under which the property of an object in [Formula: see text] can be inherited by its subobjects and quotient objects. Then, we introduce the notions of one-sided (strongly) Gorenstein subcategories. Under the assumption that [Formula: see text] is closed under countable direct sums (respectively, direct products), we prove that an object is in right (respectively, left) Gorenstein category [Formula: see text] (respectively, [Formula: see text]) if and only if it is a direct summand of an object in right (respectively, left) strongly Gorenstein subcategory [Formula: see text] (respectively, [Formula: see text]). As applications, some known results are obtained as corollaries.