Chapter 3, Completely Reducible Modules Video Solutions, A Course in Ring Theory

Problem 1

Let $\left\{V_{i} \mid i \in \mathcal{I}\right\}$ be a family of submodules of $V .$ Show that there exists a natural epimorphism $\theta: \oplus \sum_{i} V_{i} \rightarrow \sum_{i} V_{i}$ and describe $\operatorname{Ker}(\theta)$.

Jacob Fry

Numerade Educator

07:15

Problem 2

Suppose $V_{1}, V_{2}, \ldots, V_{n}$ are finitely many $R$-submodules of $V$ with $\bigcap_{1}^{n} V_{i}=0 .$ If each $V / V_{i}$ is completely reducible, prove that $V$ is also. To this end, first observe that $V$ embeds in $\oplus \sum_{1}^{n} V / V_{i}$. Now show by example that the result fails if $n$ is allowed to be infinite. Conclude therefore that a strong direct sum of completely reducible modules is not necessarily completely reducible.

Chris Trentman

Numerade Educator

02:45

Problem 3

Let $I_{1}, I_{2}, \ldots, I_{n}$ be a finite collection of two-sided ideals of $R$ such that $I_{i}+I_{j}=R$ for all $i \neq j$. If $a_{1}, a_{2}, \ldots, a_{n}$ are any elements of $R$ prove that there exists $r \in R$ with $r \equiv a_{i} \bmod I_{i}$ for all $i .$ Deduce that $R /\left(\bigcap_{1}^{n} I_{i}\right) \cong \oplus \sum_{1}^{n} R / I_{i} .$ This is the Chinese Remainder Theorem.

Trang Hoang

Numerade Educator

03:56

Problem 4

Let $p$ be a fixed prime number and let $A$ be the multiplicative group of complex $p^{n}$ th roots of unity for all $n \geq 0 .$ If we view $A$ additively as a module over the integers, show that $A$ satisfies min but that it does not have a composition series.

Sriparna Bhattacharjee

Numerade Educator

01:05

Problem 5

Let $0=V_{0} \subseteq V_{1} \subseteq \cdots \subseteq V_{n}=V$ be a series for the $R$-module $V$ and let $W \subseteq V$. Show that $W$ has a series of length $n$ whose factors are isomorphic to submodules of the factors of the $V$-series. Similarly, prove that $V / W$ has a series of length $n$ whose factors are homomornhic ima.ges of the factors of the $V$-series.

Anthony Ramos

Numerade Educator

03:13

Problem 6

Let $R \supseteq S$ be rings and assume that $R_{S}$ is a finitely generated $S$ module. If $S$ is Artinian, prove that $R$ is also. In particular, if $R=$ $\mathrm{M}_{n}(S)$ deduce that $S$ is Artinian if and only if $R$ is Artinian.

Gideon Idumah

Numerade Educator

04:07

Problem 7

Suppose $K \subseteq F$ are fields with $\operatorname{dim}_{K} F=\infty$ and let $R$ be the subring of $M_{2}(F)$ given by $R=\left(\begin{array}{cc}K & F \\ 0 & F\end{array}\right) .$ Show first that $I=\left(\begin{array}{ll}0 & F \\ 0 & 0\end{array}\right)$ is a minimal right ideal of $R$ and then construct a composition series for $R_{R} .$ Deduce that $R$ is right Artinian and then prove that $R$ is not left Artinian.

Anthony Ramos

Numerade Educator

07:44

Problem 8

Show that there is a one-to-one correspondence between idempotents $e \in \operatorname{End}_{R}(V)$ and direct sum decompositions $V=X+Y$

Anthony Ramos

Numerade Educator

02:48

Problem 9

Let $I$ be a two-sided ideal of $R$. Prove that $I=e R$ for some central idempotent $e \in R$ if and only if $R=I+J$ for some two-sided ideal $J$. When this occurs, show that $e$ and $J$ are uniquely determined by $I$.

Mohan Jain

Numerade Educator

00:44

Problem 10

Prove that all $R$-modules are free if and only if $R$ is a division ring. This is quite simple and does not require Theorem $3.9 .$

Kimberley Hoffman

Numerade Educator