91Ó°ÊÓ

If \(G\) is a finite \(p\) -primary abelian group, and if \(x \in G\) has largest order, then \(\langle x\rangle\) is a direct summand of \(G\).

(i) The intersection of any family of submodules of an \(R\) -module \(V\) is itself a submodule. (ii) If \(X\) is a subset of \(V\), let \(\langle X\rangle\) denote the submodule generated by \(X ;\) that is, \(\langle X\rangle\) is the intersection of all the submodules of \(V\) containing \(X\). If \(X \neq \varnothing\), show that \(\langle X\rangle\) is the set of all \(R\) -linear combinations of elements of \(X\); that is, $$\langle X\rangle=\left\\{\text { finite sums } \sum r_{i} x_{i}: r_{i} \in R \text { and } x_{i} \in X\right\\}$$ In particular, the cyclic submodule generated by \(v\), denoted by \(\langle v\rangle\), is \(\\{r v: r \in R\\}\)

Assuming the Basis Theorem, use the Krull-Schmidt theorem to prove the Fundamental Theorem of Finite Abelian Groups.