91Ó°ÊÓ

Use left exactness of Hom to prove that if \(G\) is an abelian group, then \(\operatorname{Hom}_{\mathbb{Z}}\left(\mathbb{I}_{n}, G\right) \cong G[n]\), where \(G[n]=\\{g \in G: n g=0\\}\).

Prove that if \(f: M \rightarrow N\) is an \(R\)-map and \(K\) is a submodule of a left \(R\)-module \(M\) with \(K \subseteq \operatorname{ker} f\), then \(f\) induces an \(R\)-map \(\widehat{f}: M / K \rightarrow N\) by \(\widehat{f}: m+K \mapsto f(m)\)

Assume that a ring \(R\) has IBN; that is, if \(R^{m} \cong R^{n}\) as left \(R\) modules, then \(m=n\). Prove that if \(R^{m} \cong R^{n}\) as right \(R\)-modules, then \(m=n\). Hint. If \(R^{m} \cong R^{n}\) as right \(R\)-modules, apply \(\operatorname{Hom}_{R}(\square, R)\), using Proposition 2.54(iii).