Remark 2

— By VII, p. 7, Remark 3, a finitely generated torsion A-module is cyclic if and only if each of its π-primary components is cyclic.

An important special case where Th. 1 and Prop. 2 apply is when $A = \mathbf{Z}$; a $\mathbf{Z}$-module is just an abelian group. An abelian group is called a torsion group if it is a torsion $\mathbf{Z}$-module, that is if all its elements have finite order. Then $P$ is taken to be the set of prime numbers $> 0$; for each prime number $p > 0$, an (abelian) group is said to be $p$-torsion if all its elements have orders which are powers of $p$. In this terminology, Th. 1 shows that every torsion abelian group is a direct sum of p-torsion groups. In the case of finite groups, this also follows from I, p. 80, Th. 4.