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A Course in Ring Theory

Donald S. Passman

Chapter 4

Wedderburn Rings - all with Video Answers

Educators

Chapter Questions

06:11

Problem 1

Show by example that the addition formulas of Lemma $4.1(\mathrm{ii})$ (iii) fail for infinite direct sums.

Subham Jyoti Mishra
Subham Jyoti Mishra
Numerade Educator
08:25

Problem 2

Let $V=\cdot \sum_{k=1}^{n} V_{k}$ be a direct sum of $R$-modules. Prove that the endomorphism ring $\operatorname{End}_{R}(V)$ is isomorphic to the checkered matrix ring
$$
S=\left\{\left(\alpha_{i, j}\right) \mid \alpha_{i, j} \in \operatorname{Hom}_{R}\left(V_{j}, V_{i}\right)\right\}
$$
Here multiplication of entries is given by function composition.

Ely Crowder
Ely Crowder
Numerade Educator
03:50

Problem 3

Assume that $R$ does not h?e IBN. Show that $M_{i}(R) \cong M_{j}(R)$ for some $i \neq j$. The converse of this is not true. Indeed, let $K$ be a field and define the rings $S_{n}$ inductively by $S_{0}=K$ and $S_{n+1}=\mathrm{M}_{2}\left(S_{n}\right) \supseteq$ $S_{n}$ for $n \geq 0$. If $S=\bigcup_{n=0}^{\infty} S_{n}$, show that $S \cong \mathrm{M}_{2}(S)$ but that $S$ has IBN.

Bobby Barnes
Bobby Barnes
University of North Texas
04:11

Problem 4

Suppose the ring $R$ has elements $e_{i, j}$ for $i, j=1,2, \ldots, n$ that satisfy $e_{i, j} e_{j^{\prime}, k}=0$ if $j \neq j^{\prime}, e_{i, j} e_{j, k}=e_{i, k}$ and $1=e_{1,1}+e_{2,2}+\cdots+e_{n, n} .$ If $S$ is the centralizer in $R$ of these $n^{2}$ elements, prove that $R \cong \mathrm{M}_{n}(S)$ and that $S \cong e_{1,1} R e_{1,1} .$ To start with, define a map $\sigma: \mathrm{M}_{n}(S) \rightarrow R$ by $\left(s_{i, j}\right) \mapsto \sum_{i, j} s_{i, j} e_{i, j} .$ Then show that $\sigma$ is a ring isomorphism.

Vishnu P
Vishnu P
Numerade Educator
02:33

Problem 5

Prove that an Artinian ring with no nonzero nilpotent elements and no nontrivial central idempotents is a division ring.

Shahab Ullah
Shahab Ullah
Numerade Educator
03:13

Problem 6

Let $R=\mathrm{M}_{n}(D)$ with $D$ a division ring. Show that $R$ has finitely many right ideals if and only if either $n=1$ or $D$ is finite.

Gideon Idumah
Gideon Idumah
Numerade Educator
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Problem 7

Let $K$ be a field and let $R \subseteq M_{2}(K)$ be given by $R=\left(\begin{array}{cc}K & K \\ 0 & K\end{array}\right)$. If $e=e_{1,1}$, show that $\operatorname{End}_{R}(e R)$ is a field even though $e R$ is not an irreducible $R$-module.

Nick Johnson
Nick Johnson
Numerade Educator
03:13

Problem 8

Write $R=\cdot \sum_{k=1}^{m} \mathrm{M}_{n_{k}}\left(D_{k}\right)$ as in the Artin-Wedderburn Theorem. If $I \triangleleft R$, show that $I$ is a direct sum of certain of the $\mathrm{M}_{n_{k}}\left(D_{k}\right) .$ In particular, deduce that the $\mathrm{M}_{n_{k}}\left(D_{k}\right)$ are the unique minimal two-sided ideals of $R$ and that, if $I \neq R$, then $R / I$ is a Wedderburn ring.

Gideon Idumah
Gideon Idumah
Numerade Educator
02:46

Problem 9

If $R=\mathrm{M}_{n}(S)$, show that all ideals of $R$ are of the form $\mathrm{M}_{n}(I)$ for $I \triangleleft S$. Furthermore, prove that $e_{i, i} R \cong e_{j, j} R$ as right $R$-modules.

Michael Jacobsen
Michael Jacobsen
Numerade Educator
07:44

Problem 10

Let $D$ be a division ring and let $R$ be a ring of $D$-linear transformations on $_{D} V$. Assume that $R$ is doubly transitive on $V$ so that, by definition, if $\left\{v_{1}, v_{2}\right\}$ is a $D$-linearly independent subset of $V$ and if $\left\{w_{1}, w_{2}\right\} \subseteq$ $V$ is arbitrary, then there exists $r \in R$ with $v_{1} r=w_{1}$ and $v_{2} r=w_{2}$. Show that $\operatorname{End}_{R}(V)=D$ and conclude that $R$ is a dense ring of linear transformations on $V .$ What happens if we merely assume that $R$ is transitive on $V ?$

Anthony Ramos
Anthony Ramos
Numerade Educator