Submodule

A submodule $N$ of an $R$-module $M$ is a subset closed under addition, additive inverses, and scalar multiplication by every element of $R$.

A submodule $N\subseteq M$ is an essential submodule if it intersects every nonzero submodule of $M$ nontrivially. A nonzero module is a uniform module precisely when every nonzero submodule is an essential submodule.

A primary decomposition expresses an ideal or submodule as an intersection of primary components.

Every proper ideal in a Noetherian ring is a finite intersection of primary ideals. This existence theorem does not give uniqueness of the individual components: embedded primary components may vary.

A proper submodule $N⊊M$ is primary when $rm\in N$ and $m\notin N$ imply $r^{k}M\subseteq N$ for some positive integer $k$.