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On the compressed essential graph of a commutative ring. (English) Zbl 1423.13031

Summary: Let \(R\) be a commutative ring. In this paper, we introduce and study the compressed essential graph of \(R, EG_E(R)\). The compressed essential graph of \(R\) is a graph whose vertices are equivalence classes of non-zero zero-divisors of \(R\) and two distinct vertices \([x]\) and \([y]\) are adjacent if and only if \(\mathrm{ann}(xy)\) is an essential ideal of \(R\). It is shown if \(R\) is reduced, then \(EG_E(R)=\Gamma_E(R)\), where \(\Gamma_E(R)\) denotes the compressed zero-divisor graph of \(R\). Furthermore, for a non-reduced Noetherian ring \(R\) with \(3<|EG_E(R)|<\infty \), it is shown that \(EG_E(R)=\Gamma_E(R)\) if and only if
(i)
\(\mathrm{Nil}(R)=\mathrm{ann}(Z(R))\).
(ii)
Every non-zero element of \(\mathrm{Nil}(R)\) is irreducible in \(Z(R)\).

MSC:

13A15 Ideals and multiplicative ideal theory in commutative rings
05C99 Graph theory
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