<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let <italic>M</italic> be a module over a commutative ring and let <italic>Max</italic>(<italic>M</italic>) be the collection of all maximal submodules of <italic>M</italic>. We topologize <italic>Max</italic>(<italic>M</italic>) with quasi-Zariski topology, where <italic>M</italic> is a Max-top module. For a subset <italic>T</italic> of <italic>Max</italic>(<italic>M</italic>), we introduce a new graph <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2018-0032_eq_001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>G</mi><mo>(</mo><msubsup><mrow><mi>τ</mi></mrow><mi>T</mi><mrow><mo>*</mo><mi>m</mi></mrow></msubsup><mo>)</mo></mrow></math><tex-math>$G(\tau_T^{*m})$</tex-math></alternatives></inline-formula>, called the quasi-Zariski topology-graph on the maximal spectrum of <italic>M</italic>. It helps us to study algebraic (resp. topological) properties of <italic>M</italic> (resp. <italic>Max</italic>(<italic>M</italic>)) by using the graphs theoretical tools.</p></abstract>

Let M be a module over a commutative ring and let Max(M) be the collection of all maximal submodules of M. We topologize Max(M) with quasi-Zariski topology, where M is a Max-top module. For a subset T of Max(M), we introduce a new graph G(τT*m)$G(\tau_T^{*m})$, called the quasi-Zariski topology-graph on the maximal spectrum of M. It helps us to study algebraic (resp. topological) properties of M (resp. Max(M)) by using the graphs theoretical tools.

The quasi-Zariski topology-graph on the maximal spectrum of modules over commutative rings

Department of Mathematics, Faculty of mathematical Sciences

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Let M be a module over a commutative ring and let Max(M) be the collection of all maximal submodules of M. We topologize Max(M) with quasi-Zariski topology,...

The quasi-Zariski topology-graph on the maximal spectrum of modules over commutative rings

Published Online: Dec 31, 2018

Page range: 41 - 56

Received: Nov 30, 2013

Accepted: Nov 27, 2017

DOI: https://doi.org/10.2478/auom-2018-0032

Keywords
prime submodule, Max-top module, quasi-Zariski topology, maximal submodule, graph, vertices, annihilating-submodule

© 2018 H. Ansari-Toroghy et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

The quasi-Zariski topology-graph on the maximal spectrum of modules over commutative rings

Published Online: Dec 31, 2018

Page range: 41 - 56

Received: Nov 30, 2013

Accepted: Nov 27, 2017

DOI: https://doi.org/10.2478/auom-2018-0032

Keywordsprime submodule, Max-top module, quasi-Zariski topology, maximal submodule, graph, vertices, annihilating-submodule

© 2018 H. Ansari-Toroghy et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
prime submodule, Max-top module, quasi-Zariski topology, maximal submodule, graph, vertices, annihilating-submodule