On the Connectedness of the Complement of a Unit Graph of a Commutative Ring
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Keywords:
Connected graph, Max(R)Abstract
The rings considered in this article are commutative with identity. We denote the set of all maximal ideals of a ring $R$ by $Max(R)$ and we denote the Jacobson radical of $R$ by $J(R)$. Let $R$ be a ring. Recall from [2] that the {\it unit graph} of $R$, denoted by $G(R)$, is an undirected graph whose vertex set of all elements of $R$ and distinct vertices $x, y$ are joined by an edge in this graph if and only if $x + y\in U(R)$. In this article, we studied Complement of unit graph and we denoted it $(UG(R))^{c}$. Hence, in this graph two elements $x, y$ are joined by an edge in $(UG(R))^{c}$ if and only if $ x + y \in NU(R)$. In this article we proved some results on connectedness of $(UG(R))^{c}$.
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