Submodule

A submodule $N$ of a 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.

An essential right ideal is an essential submodule of the right regular module of a ring. Every right ideal containing an essential right ideal is essential.

If $c$ is a regular element of a ring in a right Noetherian ring and a nonzero right ideal $B$ had $B\cap cR=0$, cancellation would make $B+cB+c^{2}B+\cdots$ an infinite direct sum of nonzero right ideals. Its finite partial sums contradict the ascending chain condition. Thus $cR$ is an essential right ideal.

An extension $N\subseteq M$ is essential when $N$ is an essential submodule of $M$. Essentiality is transitive. It is an intersection condition, not a statement that the quotient is small in dimension or finitely generated.

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 $k$.