Some results on the annihilators and attached primes of local cohomology modules

Let $(R,\mathfrak{m})$ be a local ring and $M$ a finitely generated $R$-module. It is shown that if $M$ is relative Cohen-Macaulay with respect to an ideal $\mathfrak{a}$ of $R$, then $$\text{Ann}$R(H{\mathfrak{a}}^{$\text{cd}$(\mathfrak{a}, M)}(M))=$\text{Ann}$_RM/L=$\text{Ann}$_RM$and$$\text{Ass}$_R(R/$\text{Ann}$_RM)\subseteq {\mathfrak{p} \in $\text{Ass}$R M|,{\rm cd}(\mathfrak{a}, R/\mathfrak{p})=$\text{cd}$(\mathfrak{a}, M)},$where$L$isthelargestsubmoduleof$M$suchthat${\rm cd}(\mathfrak{a}, L)< {\rm cd}(\mathfrak{a}, M)$\mathrm{.}Wealsoshowthatif$H^{\dim M}{\mathfrak{a}}(M)=0$, then $$\text{Att}$R(H^{\dim M-1}{\mathfrak{a}}(M))= {\mathfrak{p} \in $\text{Supp}$ (M)|,{\rm cd}(\mathfrak{a}, R/\mathfrak{p})=\dim M-1},$ and so the attached primes of $H_{\mathfrak{a}}^{\dim M-1}(M)$ depends only on $\text{Supp}(M)$. Finally, we prove that if $M$ is an arbitrary module (not necessarily finitely generated) over a Noetherian ring $R$ with ${\rm cd}(\mathfrak{a}, M)={\rm cd}(\mathfrak{a}, R/$\text{Ann}$_RM)$,then$$\text{Att}$R(H^{{\rm cd}(\mathfrak{a}, M)}{\mathfrak{a}}(M))\subseteq{\mathfrak{p} \in V($\text{Ann}$_RM)|,{\rm cd}(\mathfrak{a}, R/\mathfrak{p})={\rm cd}(\mathfrak{a}, M)}.$Asaconsequenceofthisitisshownthatif$\dim M=\dim R$,then$$\text{Att}$R(H^{\dim M}{\mathfrak{a}}(M))\subseteq{\mathfrak{p} \in $\text{Ass}$_R M|,{\rm cd}(\mathfrak{a}, R/\mathfrak{p})=\dim M}.$