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Let $(R,𝔪)$ be a local ring, $𝔞$ an ideal of $R$ and $M$ a nonzero Artinian $R$-module of Noetherian dimension $n$ with $\mathrm{hd}(𝔞,M)=n$. We determine the annihilator of the top local homology module ${\mathrm{H}}_n^{𝔞}\left(M\right)$. In fact, we prove that $${\mathrm{Ann}}_R\left({\mathrm{H}}_n^{𝔞}\left(M\right)\right)={\mathrm{Ann}}_R\left(N(𝔞,M)\right),$$ where $N(𝔞,M)$ denotes the smallest submodule of $M$ such that $\mathrm{hd}(𝔞,M/N(𝔞,M\left)\right)<n$. As a consequence, it follows that for a complete local ring $(R,𝔪)$ all associated primes of ${\mathrm{H}}_n^{𝔞}\left(M\right)$ are minimal.

Let $𝔞$ be an ideal of Noetherian local ring $(R,𝔪)$ and $M$ a finitely generated $R$-module of dimension $d$. In this paper we investigate the Artinianness of formal local cohomology modules under certain conditions on the local cohomology modules with respect to $𝔪$. Also we prove that for an arbitrary local ring $(R,𝔪)$ (not necessarily complete), we have ${\mathrm{Att}}_R\left({𝔉}_{𝔞}^d\left(M\right)\right)=\mathrm{Min}\mathrm{V}\left({\mathrm{Ann}}_R{𝔉}_{𝔞}^d\left(M\right)\right).$

Let $I$ be an ideal of Noetherian ring $R$ and $M$ a finitely generated $R$-module. In this paper, we introduce the concept of weakly colaskerian modules and by using this concept, we give some vanishing and finiteness results for local homology modules. Let $I_M:={Ann}_R(M/IM)$, we will prove that for any integer $n$ If ...