<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let <italic>R</italic> be a commutative ring with identity which is not an integral domain. An ideal <italic>I</italic> of a ring <italic>R</italic> is called an annihilating ideal if there exists <italic>r</italic> ∈ <italic>R −</italic> {0} such that <italic>Ir</italic> = (0). Visweswaran and H. D. Patel associated a graph with the set of all non-zero annihilating ideals of <italic>R</italic>, denoted by Ω(<italic>R</italic>) as the graph with the vertex-set <italic>A</italic>(<italic>R</italic>)<sup>*</sup>, the set of all non-zero annihilating ideals of <italic>R</italic> and two distinct vertices <italic>I, J</italic> are joined if and only if <italic>I</italic> +<italic>J</italic> is also an annihilating ideal of <italic>R</italic>. In this paper, we study the metric dimension of Ω(<italic>R</italic>) and provide metric dimension formulas for Ω(<italic>R</italic>).</p></abstract>

Let R be a commutative ring with identity which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r ∈ R − {0} such that Ir = (0). Visweswaran and H. D. Patel associated a graph with the set of all non-zero annihilating ideals of R, denoted by Ω(R) as the graph with the vertex-set A(R)*, the set of all non-zero annihilating ideals of R and two distinct vertices I, J are joined if and only if I +J is also an annihilating ideal of R. In this paper, we study the metric dimension of Ω(R) and provide metric dimension formulas for Ω(R).

On the metric dimension of a total graph of non-zero annihilating ideals

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

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{"article-title":"On the metric dimension of a total graph of non-zero annihilating ideals"}

Let R be a commutative ring with identity which is not an integral domain. An ideal I of a ring R is called an annihilating ideal if there exists r ∈ R − {0}...

On the metric dimension of a total graph of non-zero annihilating ideals

Published Online: Dec 28, 2020

Page range: 5 - 14

Received: Sep 16, 2019

Accepted: Jan 09, 2020

DOI: https://doi.org/10.2478/auom-2020-0031

Keywords
metric dimension, Zero-divisor, Annihilating ideal, Resolving set

© 2020 Nazi Abachi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

On the metric dimension of a total graph of non-zero annihilating ideals

Published Online: Dec 28, 2020

Page range: 5 - 14

Received: Sep 16, 2019

Accepted: Jan 09, 2020

DOI: https://doi.org/10.2478/auom-2020-0031

Keywordsmetric dimension, Zero-divisor, Annihilating ideal, Resolving set

© 2020 Nazi Abachi et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Keywords
metric dimension, Zero-divisor, Annihilating ideal, Resolving set