The forcing metric dimension of a total graph of non-zero annihilating ideals
UDC 519.17 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 $r\in R- \{0\}$ such that $Ir=(0)$.  The  total graph of non-zero annihilating idea...
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| Date: | 2023 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2023
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7011 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | UDC 519.17
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 $r\in R- \{0\}$ such that $Ir=(0)$.  The  total graph of non-zero annihilating ideals of $R,$ denoted by $\Omega(R),$ is а graph with the vertex set $A(R)^*,$ the set of all non-zero annihilating ideals of $R,$ and two distinct vertices $I$ and $J$ are joined  if and only if  $I+J$ is also an  annihilating ideal of $R$. We study the forcing metric dimension of $\Omega(R)$ and determine the forcing metric dimension of  $\Omega(R)$.  It is shown that the forcing metric dimension of  $\Omega(R)$ is equal either to zero or to the metric dimension. |
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| DOI: | 10.37863/umzh.v75i6.7011 |