Uniform module

Remarks

Equivalently, $M \ne 0$ and every nonzero submodule of $M$ is essential in $M$. Example: the $\Z$-module $\Z$ is uniform. The check: nonzero submodules $a\Z$ and $b\Z$ both contain $ab \ne 0$. Example: $\Z/p^n\Z$, for $p$ prime and $n \ge 1$, is uniform, since its subgroups form a chain. Non-example: $\Z \oplus \Z$ fails clause 2, since $\Z \oplus 0$ and $0 \oplus \Z$ meet only in $0$. Every uniform module is indecomposable. The converse fails: let $K$ be a field and $A = K[x, y]/(x^2, xy, y^2)$. As a module over itself, $A$ is indecomposable, because $A$ is a commutative local ring and so has no idempotents other than $0$ and $1$. It is not uniform, since its submodules $Ax = Kx$ and $Ay = Ky$ meet only in $0$.