Chapter 1, Modules and Homomorphisms Video Solutions, A Course in Ring Theory

Problem 1

Suppose $V$ is an additive abelian group and that there exists a multiplicative map $V \times R \rightarrow V$ satisfying conditions (i), (ii), and (iii) of the definition of an $R$-module. Prove that $V=V_{0}+V_{1}$, where $V_{0}$ and $V_{1}$ are the additive subgroups given by $V_{0}=\{v \in V \mid v 1=0\}$ and $V_{1}=\{v \in V \mid v 1=v\}$. Deduce that $V_{0} R=0$ and that $V_{1}$ is a (unital) $R$-module.

Ely Crowder

Numerade Educator

01:02

Problem 2

Let $\theta: G \rightarrow H$ be a homomorphism of groups. Prove that $\theta$ sends the identity of $G$ to the identity of $H$ and that it sends inverses to inverses.

Varsha Aggarwal

Numerade Educator

08:07

Problem 3

If $V$ is a right $R$-module, prove that $v 0=0=0 r$ for all $v \in V$ and $r \in R$. Furthermore, show that $(-v) r=-(v r)=v(-r)$.

William Mead

Numerade Educator

03:58

Problem 4

Let $W$ be a subset of $V$. Prove that $W$ is a submodule if and only if it is nonempty, closed under $+$ and closed under multiplication by $R$. If $X$ and $Y$ are submodules of $V$, show that $X \cap Y$ and $X+Y$ are also submodules.

Anthony Ramos

Numerade Educator

00:59

Problem 5

Let $V$ and $W$ be additive abelian groups. Verify that $\operatorname{Hom}(V, W)$ is an additive abelian group and that $\operatorname{End}(V)$ is a ring. If $V$ and $W$ are $R$-modules, verify that $\operatorname{Hom}_{R}(V, W)$ is a subgroup of $\operatorname{Hom}(V, W)$ and that $\operatorname{End}_{R}(V)$ is a subring of $\operatorname{End}(V)$.

Varsha Aggarwal

Numerade Educator

Problem 6

If $V$ and $W$ are vector spaces over the field $K$, observe that they are $K$-modules and describe $\operatorname{Hom}_{K}(V, W)$ and $\operatorname{End}_{K}(V)$. Be more specific in case $\operatorname{dim}_{K} V=n<\infty$ and $\operatorname{dim}_{K} W=m<\infty$.

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01:02

Problem 7

Show that the diagram
$$
\begin{array}{ccccc}
A & \stackrel{\sigma}{\longrightarrow} & B & \stackrel{\tau}{\longrightarrow} & C \\
\alpha \downarrow & \downarrow \beta & & \downarrow \gamma \\
A^{\prime} & \stackrel{\sigma^{\prime}}{\longrightarrow} & B^{\prime} & \stackrel{\tau^{\prime}}{\longrightarrow} & C^{\prime}
\end{array}
$$

Raj Bala

Numerade Educator

01:42

Problem 8

Suppose
$\begin{aligned} A & \stackrel{\sigma}{\longrightarrow} \quad B \\ \alpha \downarrow & \downarrow \beta \\ A^{\prime} & \stackrel{\sigma^{\prime}}{\longrightarrow} \quad B^{\prime} \end{aligned}$
is a commutative diagram of $R$-modules and of $R$-homomorphisms. Prove that $\alpha: \operatorname{Ker}(\sigma) \rightarrow \operatorname{Ker}\left(\sigma^{\prime}\right)$ and furthermore that $\beta$ induces a $\operatorname{map} \tilde{\beta}: B / \operatorname{Im}(\sigma) \rightarrow B^{\prime} / \operatorname{Im}\left(\sigma^{\prime}\right) .$ Here $B / \operatorname{Im}(\sigma)$ is called the cokernel of $\sigma$.

Urvashi Arora

Numerade Educator

04:07

Problem 9

Let $V$ be a nonzero finitely generated $R$-module. Prove that $V$ has a maximal submodule $W$. By this we mean that $W \neq V$ and that there are no submodules contained properly between $W$ and $V$. Show by example that this result is false for arbitrary nonzero $V$.

Anthony Ramos

Numerade Educator

09:31

Problem 10

If the collection of subspaces of the $K$-vector space $V$ satisfies either distributive law $A+(B \cap C)=(A+B) \cap(A+C)$ or $A \cap(B+C)=$ $(A \cap B)+(A \cap C)$, show that $\operatorname{dim}_{K} V \leq 1$

Chris Trentman

Numerade Educator