Example 2

Let M be an A-module. If there exists an infinite family $(M_i)_{i\in I}$ of nonzero submodules of M whose sum is direct, then M is neither Artinian nor Noetherian: indeed, for every strictly decreasing (resp. strictly increasing) infinite sequence $(J_n)$ of subsets of I, the infinite sequence $(\sum_{i\in J_n}M_i)$ of submodules of M is strictly decreasing (resp. strictly increasing). In particular, an infinite-dimensional vector space over a field is neither Artinian nor Noetherian.