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

M.  Pazoki

doi:10.37863/umzh.v75i6.7011

M. Pazoki Department of Mathematics, Parand Branch, Islamic Azad University, Iran

DOI: https://doi.org/10.37863/umzh.v75i6.7011

Keywords: forcing metric dimension Zero-divisor; Annihilating ideal, Resolving set

Abstract

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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