Lynch’s conjecture and annihilators of local cohomology modules

Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$ with $\cd(I,R)=t\geq 1$. In this paper, we consider the Lynch's conjecture and we obtain a partial answer for this conjecture. More precisely, we show that if $M$ is an $R$-module such that $0\neq H^t_I(M)$ is $I$-cofinite, then $\Ann_RH^t_I(R)\subseteq \p$ for some minimal prime ideal $\p$ of $R$.