On the Attached Primes of Top Local Cohomology Modules

Let \(\mathfrak {a}\) $\mathfrak{a}$ be an ideal of a Noetherian ring R and M be a finitely generated R-module with \(\textrm{cd}(\mathfrak {a},M)=c\ge 1\) $\text{cd}(\mathfrak{a},M)=c\ge 1$ . In this paper, we prove that \( \textrm{mAtt}_R\,H^c_{\mathfrak {a}}(M) \subseteq \textrm{mAss}_R\,M \cup \{\mathfrak {p}\in \textrm{Supp}\,M: \textrm{Ann}_R\,H^{c-1}_{\mathfrak {a}}(R/\mathfrak {p})=\mathfrak {p}=\textrm{Ann}_R\,H^{c}_{\mathfrak {a}}(R/\mathfrak {p})\}. \) $${\text{mAtt}}_R{H}_{\mathfrak{a}}^c(M)\subseteq {\text{mAss}}_{R}M\cup \{\mathfrak{p}\in \text{Supp}M:{\text{Ann}}_R{H}_{\mathfrak{a}}^{c-1}(R/\mathfrak{p})=\mathfrak{p}={\text{Ann}}_R{H}_{\mathfrak{a}}^c(R/\mathfrak{p})\}.$$ Moreover, we show that \( \textrm{Att}_R\,H^c_{\mathfrak {a}}(M)\subseteq \textrm{mAss}_R\,M \cup \{\mathfrak {p}\in \textrm{Supp}\,M: \textrm{Ann}_R\,H^{c-1}_{\mathfrak {a}}(R/\mathfrak {p})=\mathfrak {p}=\textrm{Ann}_R\,H^{c}_{\mathfrak {a}}(R/\mathfrak {p})\}, \) $${\text{Att}}_R{H}_{\mathfrak{a}}^c(M)\subseteq {\text{mAss}}_{R}M\cup \{\mathfrak{p}\in \text{Supp}M:{\text{Ann}}_R{H}_{\mathfrak{a}}^{c-1}(R/\mathfrak{p})=\mathfrak{p}={\text{Ann}}_R{H}_{\mathfrak{a}}^c(R/\mathfrak{p})\},$$ whenever the R-module \(H^c_{\mathfrak {a}}(M)\) $H_{\mathfrak{a}}^c(M)$ is Artinian.