Example 4

Let M be an A-module that is the direct sum of an infinite family $(M_i)_{i\in I}$ of nonzero submodules. Let E be the endomorphism ring of M. For every $i\in I$, let $\mathfrak{a}_i$ (resp. $\mathfrak{b}_i$) be the set of elements of E with kernel containing $\sum_{j\not=i}M_j$ (resp. with image contained in $M_i$). Then $(\mathfrak{a}_i)$ is an infinite family of nonzero left ideals of E whose sum is direct, and $(\mathfrak{b}_i)$ is an infinite family of nonzero right ideals of E whose sum is direct. Consequently, the ring E is neither left nor right Artinian (resp. Noetherian) (VIII, p. 2, Example 2). In particular, the endomorphism ring of an infinite-dimensional vector space is neither left nor right Artinian (resp. Noetherian).