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On the strong metric dimension of a total graph of nonzero annihilating ideals. (English) Zbl 1486.13013

Summary: Let \(R\) be a commutative ring with identity which is not an integral domain. An ideal \(I\) of \(R\) is called an annihilating ideal if there exists \(r\in R- \{0\}\) such that \(Ir=(0)\). The total graph of nonzero annihilating ideals of \(R\) is the graph \(\Omega (R)\) whose vertices are the nonzero annihilating ideals of \(R\) and two distinct vertices \(I,J\) are joined if and only if \(I+J\) is also an annihilating ideal of \(R\). We study the strong metric dimension of \(\Omega (R)\) and evaluate it in several cases.

MSC:

13A70 General commutative ring theory and combinatorics (zero-divisor graphs, annihilating-ideal graphs, etc.)
13A15 Ideals and multiplicative ideal theory in commutative rings
05C99 Graph theory
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
13B99 Commutative ring extensions and related topics
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