On the vanishing, artinianness and finiteness of local cohomology modules

Let $R$ be a noetherian ring, $\text{fa}$ an ideal of $R$, and $M$ an $R$–module. We prove that for a finite module $M$, if ${\text{LC}}_{\text{fa}}^i(M)$ is minimax for all $i\ge r\ge 1$, then ${\text{LC}}_{\text{fa}}^i(M)$ is artinian for $i\ge r$. A Local-global Principle for minimax local cohomology modules is shown. If ${\text{LC}}_{\text{fa}}^i(M)$ is coatomic for $i\le r$ ($M$ finite) then ${\text{LC}}_{\text{fa}}^i(M)$ is finite for $i\le r$. We give conditions for a module, which is locally minimax to be a minimax module. A non-vanishing theorem and some vanishing theorems are proved for local cohomology modules.