The forcing metric dimension of a total graph of non-zero annihilating ideals

• Дата: 2023
• Автор: Pazoki, M.
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Institute of Mathematics, NAS of Ukraine 2023
• Онлайн доступ: https://umj.imath.kiev.ua/index.php/umj/article/view/7011
• Назва журналу: Ukrains’kyi Matematychnyi Zhurnal

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.