On a generalization of the exchange property to modules with semilocal endomorphism rings

AbstractIt is shown that, over any ring R, the direct sum M = ⊕i∈IMi of uniform right R-modules Mi with local endomorphism rings is a CS-module if and only if every uniform submodule of M is essential in a direct summand of M and there does not exist an infinite sequence of non-isomorphic monomorphisms , with distinct in ∈ I. As a consequence, any CS-module which is a direct sum of submodules with local endomorphism rings has the exchange property.

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On Hermitian endomorphism rings

Journal of Algebra and Its Applications ◽

10.1142/s0219498817502206 ◽

2017 ◽

Vol 16 (11) ◽

pp. 1750220

Author(s):

Wanru Zhang

Keyword(s):

Endomorphism Ring ◽

Necessary Conditions ◽

Exchange Property ◽

Projective Modules ◽

Endomorphism Rings ◽

Dual Problems ◽

Injective Modules ◽

Sufficient And Necessary Conditions

Let [Formula: see text] be a right [Formula: see text]-module with finite exchange property and let [Formula: see text] be its endomorphism ring. In this paper, some sufficient and necessary conditions for [Formula: see text] to be a Hermitian ring are given. Moreover, we investigate Hermitian endomorphism rings of quasi-projective modules by means of completions of diagrams. The dual problems for quasi-injective modules are also studied.

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Power graphs and exchange property for resolving sets

Open Mathematics ◽

10.1515/math-2019-0093 ◽

2019 ◽

Vol 17 (1) ◽

pp. 1303-1309 ◽

Cited By ~ 1

Author(s):

Ghulam Abbas ◽

Usman Ali ◽

Mobeen Munir ◽

Syed Ahtsham Ul Haq Bokhary ◽

Shin Min Kang

Keyword(s):

Present Article ◽

Linear Algebra ◽

Finite Groups ◽

Robot Navigation ◽

Simple Graph ◽

Exchange Property ◽

Metric Dimension ◽

Sufficient Condition ◽

Resolving Set ◽

Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

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