Let $R$ be a commutative Noetherian ring and $a$ an ideal of $R$. Let $A$ be an Artinian $R$- module. We introduce the notion of the $n$th Artinianness dimension of $A$ relative to $a$, $g_n^a(A)=inf\{g^{a R_{p}}(^p A): p\in Spec(R) \ \ \text{and} \ \ dim R/p\geq n\}$, for all $n\in\mathbb{N}_{0}$ and prove that $g_1^a(A)=inf\{i\in\mathbb{N}_0: H_i^a(A) \ \ \text{is not minimax}\}$, whenever $R$ is a complete semi-local ring. Moreover, in this situation we show that $Coass_R(H_{g_1^a(A)}^a(A))$ is a finite set.
