Module

An $R$-module is an abelian group equipped with scalar multiplication by a ring $R$, satisfying the usual distributive and associative laws.

A torsion module over an integral domain is a module in which every element is annihilated by some nonzero scalar.

For a module $M$ over an integral domain $R$, its torsion submodule isThe domain condition makes this a submodule: products of nonzero annihilators remain nonzero and annihilate sums.

An $R$-module homomorphism is an additive map $f:M\to N$ satisfying $f(rm)=rf(m)$.

The endomorphism ring ${\mathrm{End}}_R(M)$ consists of the $R$-module homomorphisms $M\to M$, with pointwise addition and composition as multiplication.

A nonzero module is a brick when every endomorphism is either zero or an isomorphism. Equivalently, its endomorphism ring is a division ring.

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

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 $k$.