Chapter 13

Show that if \(N\) is a submodule of an \(R\) -module \(M,\) then a set \(P \subseteq N\) is a submodule of \(M\) if and only if \(P\) is a submodule of \(N\).

Let \(M_{1}\) and \(M_{2}\) be \(R\) -modules, and let \(N_{1}\) be a submodule of \(M_{1}\) and \(N_{2}\) a submodule of \(M_{2}\). Show that \(N_{1} \times N_{2}\) is a submodule of \(M_{1} \times M_{2}\)

Show that if \(R\) is non-trivial, then the \(R\) -module \(R[X]\) is not finitely generated.

Verify that the "is isomorphic to" relation on \(R\) -modules is an equivalence relation; that is, for all \(R\) -modules \(M_{1}, M_{2}, M_{3},\) we have: (a) \(M_{1} \cong M_{1}\) (b) \(M_{1} \cong M_{2}\) implies \(M_{2} \cong M_{1}\) (c) \(M_{1} \cong M_{2}\) and \(M_{2} \cong M_{3}\) implies \(M_{1} \cong M_{3}\).

EXERCISE 13.5. Let \(\rho_{i}: M_{i} \rightarrow M_{i}^{\prime},\) for \(i=1, \ldots, k,\) be \(R\) -linear maps. Show that the map $$\begin{aligned} \rho: & \boldsymbol{M}_{1} \times \cdots \times \boldsymbol{M}_{k} \rightarrow \boldsymbol{M}_{1}^{\prime} \times \cdots \times \boldsymbol{M}_{k}^{\prime} \\\ \left(\alpha_{1}, \ldots, \alpha_{k}\right) & \mapsto\left(\rho_{1}\left(\alpha_{1}\right), \ldots, \rho_{k}\left(\alpha_{k}\right)\right) \end{aligned}$$ is an \(R\) -linear map.

Let \(\rho: M \rightarrow M^{\prime}\) be an \(R\) -linear map, and let \(c \in R\). Show that \(\rho(c \boldsymbol{M})=c \rho(M)\).

Let \(\rho: M \rightarrow M^{\prime}\) be an \(R\) -linear map. Let \(N\) be a submodule of \(M,\) and let \(\tau: N \rightarrow M^{\prime}\) be the restriction of \(\rho\) to \(N\). Show that \(\tau\) is an \(R\) -linear map and that \(\operatorname{Ker} \tau=\operatorname{Ker} \rho \cap N\).

Suppose \(M_{1}, \ldots, M_{k}\) are \(R\) -modules. Show that for each \(i=\) \(1, \ldots, k,\) the projection map \(\pi_{i}: M_{1} \times \cdots \times M_{k} \rightarrow M_{i}\) that sends \(\left(\alpha_{1}, \ldots, \alpha_{k}\right)\) to \(\alpha_{i}\) is a surjective \(R\) -linear map.

Show that if \(M=M_{1} \times M_{2}\) for \(R\) -modules \(M_{1}\) and \(M_{2},\) and \(N_{1}\) is a subgroup of \(M_{1}\) and \(N_{2}\) is a subgroup of \(M_{2},\) then we have an \(R\) -module isomorphism \(M /\left(N_{1} \times N_{2}\right) \cong M_{1} / N_{1} \times M_{2} / N_{2}\)

Let \(M\) be an \(R\) -module with submodules \(N_{1}\) and \(N_{2}\). Show that we have an \(R\) -module isomorphism \(\left(N_{1}+N_{2}\right) / N_{2} \cong N_{1} /\left(N_{1} \cap N_{2}\right)\).

Let \(M\) be an \(R\) -module with submodules \(N_{1}, N_{2},\) and \(A,\) where \(N_{2} \subseteq N_{1}\). Show that \(\left(N_{1} \cap A\right) /\left(N_{2} \cap A\right)\) is isomorphic to a submodule of \(N_{1} / N_{2}\).

Let \(\rho: M \rightarrow M^{\prime}\) be an \(R\) -linear map with kernel \(K .\) Let \(N\) be a submodule of \(M\). Show that we have an \(R\) -module isomorphism \(M /(N+K) \cong\) \(\rho(\boldsymbol{M}) / \rho(N)\).

Let \(\rho: M \rightarrow M^{\prime}\) be a surjective \(R\) -linear map. Let \(S\) be the set of all submodules of \(M\) that contain \(\operatorname{Ker} \rho,\) and let \(S^{\prime}\) be the set of all submodules of \(M^{\prime} .\) Show that the sets \(S\) and \(\mathcal{S}^{\prime}\) are in one-to-one correspondence, via the map that sends \(N \in S\) to \(\rho(N) \in \mathcal{S}^{\prime}\).

Let \(M\) be an \(R\) -module. Suppose \(\left\\{\alpha_{i}\right\\}_{i=1}^{n}\) is a linearly independent family of elements of \(M\). Show that for every \(J \subseteq\\{1, \ldots, n\\},\) the subfamily \(\left\\{\alpha_{j}\right\\}_{j \in J}\) is also linearly independent.

Suppose \(\rho: M \rightarrow M^{\prime}\) is an \(R\) -linear map. Show that if \(\left\\{\alpha_{i}\right\\}_{i=1}^{n}\) is a linearly dependent family of elements of \(M,\) then \(\left\\{\rho\left(\alpha_{i}\right)\right\\}_{i=1}^{n}\) is also linearly dependent.

Suppose \(\rho: M \rightarrow M^{\prime}\) is an injective \(R\) -linear map and that \(\left\\{\alpha_{i}\right\\}_{i=1}^{n}\) is a linearly independent family of elements of \(M .\) Show that \(\left\\{\rho\left(\alpha_{i}\right)\right\\}_{i=1}^{n}\) is linearly independent.

Suppose that \(\left\\{\alpha_{i}\right\\}_{i=1}^{n}\) spans an \(R\) -module \(M\) and that \(\rho: M \rightarrow\) \(M^{\prime}\) is an \(R\) -linear map. Show that: (a) \(\rho\) is surjective if and only if \(\left\\{\rho\left(\alpha_{i}\right)\right\\}_{i=1}^{n}\) spans \(M^{\prime} ;\) (b) if \(\left\\{\rho\left(\alpha_{i}\right)\right\\}_{i=1}^{n}\) is linearly independent, then \(\rho\) is injective.

Show that if \(V_{1}, \ldots, V_{n}\) are finite dimensional vector spaces over \(F\), then \(V_{1} \times \cdots \times V_{n}\) has dimension \(\sum_{i=1}^{n} \operatorname{dim}_{F}\left(V_{i}\right)\).

Show that if \(V\) is a finite dimensional vector space over \(F\) with subspaces \(W_{1}\) and \(W_{2},\) then $$\operatorname{dim}_{F}\left(W_{1}+W_{2}\right)=\operatorname{dim}_{F}\left(W_{1}\right)+\operatorname{dim}_{F}\left(W_{2}\right)-\operatorname{dim}_{F}\left(W_{1} \cap W_{2}\right)$$.

From the previous exercise, one might be tempted to think that a more general "inclusion/exclusion principle" for dimension holds. Determine if the following statement is true or false: if \(V\) is a finite dimensional vector space over \(F\) with subspaces \(W_{1}, W_{2},\) and \(W_{3},\) then $$\begin{aligned} \operatorname{dim}_{F}\left(W_{1}\right.&\left.+W_{2}+W_{3}\right)=\operatorname{dim}_{F}\left(W_{1}\right)+\operatorname{dim}_{F}\left(W_{2}\right)+\operatorname{dim}_{F}\left(W_{3}\right) \\\ &-\operatorname{dim}_{F}\left(W_{1} \cap W_{2}\right)-\operatorname{dim}_{F}\left(W_{1} \cap W_{3}\right)-\operatorname{dim}_{F}\left(W_{2} \cap W_{3}\right) \\ &+\operatorname{dim}_{F}\left(W_{1} \cap W_{2} \cap W_{3}\right) \end{aligned}.$$

Suppose that \(V\) and \(W\) are vector spaces over \(F, V\) is finite dimensional, and \(\left\\{\alpha_{i}\right\\}_{i=1}^{k}\) is a linearly independent family of elements of \(V\). In addition, let \(\beta_{1}, \ldots, \beta_{k}\) be arbitrary elements of \(W\). Show that there exists an \(F\) linear map \(\rho: V \rightarrow W\) such that \(\rho\left(\alpha_{i}\right)=\beta_{i}\) for \(i=1, \ldots, k\).

Let \(V\) be a vector space over \(F\) with basis \(\left\\{\alpha_{i}\right\\}_{i=1}^{n} .\) Let \(S\) be a finite, non-empty subset of \(F,\) and define $$B:=\left\\{\sum_{i=1}^{n} c_{i} \alpha_{i}: c_{1}, \ldots, c_{n} \in S\right\\}$$ Show that if \(W\) is a subspace of \(V\), with \(W \subsetneq V\), then \(|B \cap W| \leq|S|^{n-1}\).