91Ó°ÊÓ

Show that any free module over an integral domain is torsion-free.

Let \(M\) be a finitely generated torsion module over a principal ideal domain. Prove that the following are equivalent: a) \(M\) is indecomposable b) \(M\) has only one elementary divisor (including multiplicity) c) \(M\) is cyclic of prime power order.

Suppose that \(M\) is a free module of finite rank over a principal ideal domain \(R\). Let \(N\) be a submodule of \(M\). If \(M / N\) is torsion, prove that \(\operatorname{rk}(N)=\operatorname{rk}(M)\).

Show that the rational numbers \(\mathbb{Q}\) form a torsion-free \(\mathbb{Z}\)-module that is not free.

Let \(R\) be a principal ideal domain and let \(M\) be a free \(R\)-module. a) Prove that a submodule \(N\) of \(M\) is complemented if and only if \(M / N\) is free. b) If \(M\) is also finitely generated, prove that \(N\) is complemented if and only if \(M / N\) is torsion-free.

Let \(M\) be a free module of finite rank over a principal ideal domain \(R\). a) Prove that if \(N\) is a complemented submodule of \(M\), then \(\operatorname{rk}(N)=\operatorname{rk}(M)\) if and only if \(N=M\). b) Show that this need not hold if \(N\) is not complemented. c) Prove that \(N\) is complemented if and only if any basis for \(N\) can be extended to a basis for \(M\).

Let \(M\) be a free module of finite rank over a principal ideal domain \(R\). Let \(L\) and \(N\) be submodules of \(M\) with \(L\) complemented in \(M\). Prove that $$ \operatorname{rk}(L+N)+\operatorname{rk}(L \cap N)=\operatorname{rk}(L)+\operatorname{rk}(N) $$