Attached primes and annihilators of top local cohomology modules defined by a pair of ideals

Assume that $R$ is a complete Noetherian local ring and $M$ is a non-zero finitely generated $R$-module of dimension $n=\dim(M)\geq 1$. It is shown that any non-empty subset $T$ of $\mathrm{Assh}(M)$ can be expressed as the set of attached primes of the top local cohomology modules $H_{I,J}^n(M)$ for some proper ideals $I,J$ of $R$. Moreover, for ideals $I, J=\bigcap_ {\mathfrak p\in \mathrm{Att}_R(H_{I}^n(M))}\mathfrak p$ and $J'$ of $R$ it is proved that $T=\mathrm{Att}_R(H_{I,J}^n(M))=\mathrm{Att}_R(H_{I,J'}^n(M))$ if and only if $J'\subseteq J$. Let $H_{I,J}^n(M)\neq 0$. It is shown that there exists $Q\in \mathrm{Supp}(M)$ such that $\dim(R/Q)=1$ and $H_Q^n(R/{\mathfrak p})\neq 0$, for each $\mathfrak p \in \mathrm{Att}_R(H_{I,J}^n(M))$. In addition, we prove that if $I$ and $J$ are two proper ideals of a Noetherian local ring $R$, then $\mathrm{Ann}_R(H_{I,J}^{n}(M))=\mathrm{Ann}_R(M/{T_R(I,J,M)})$, where $T_R(I,J,M)$ is the largest submodule of $M$ with $\mathrm{cd}(I,J,T_R(I,J,M))<\mathrm{cd}(I,J,M)$, here $\mathrm{cd}(I,J,M)$ is the cohomological dimension of $M$ with respect to $I$ and $J$. This result is a generalization of [1, Theorem 2.3] and [2, Theorem 2.6].