First Exercise

Antiautomorphism

1. Antiautomorphism

Let $R$ be a ring. Show that $R\cong R^{op}$ if and only if there is exists antiautormorphism $\alpha :R\to R$. (An antiautomorphism is a bijection which preserves addition and 1, but reverses multiplication.)

2. Matrix Ring with commutative ring

Show that if $K$ is a commutative ring, that $M_{n}K\cong {\left(M_{n}K\right)}^{op}$.

3. Real Quaternion

Show that $\mathbb{H}\cong {\mathbb{H}}^{op}$ where $\mathbb{H}$ is the real quaternions.

Group ring has zero divisors.

Property

Module

1. Show that if $M$ is a proper maximal submodule of an $R$-module $F$, that $F/M$ is simple.
2. Find a submodule $F$ over $\mathbb{Z}$ which has no maximal proper submodule.
3. Show that finitely generated $R$-module has a maximal proper submodules.

Simple Module

1. Show that ${\mathbb{R}}^2$ is a simple $M_2\left(\mathbb{R}\right)$-module.
2. Express $M_2\left(\mathbb{R}\right)$ as a direct sum of simple modules.

Central Idempotent

Let $R$ be a ring. Then $R\cong R_1\times R_2$ with $R_i$ nontrivial if and only if $R$ has a nontrivial central idempotent.

Burnside Theorem

Let $D_{2n}=⟨r,s|r^n=s^2=sr{s}^{-1}r⟩$ be the dihedral group of order $2n$, which is the isometry group of the regular $n$-gon. Let ${\xi }_n=e^{2\pi i/n}\in \mathbb{C}$, which is a primitivie $n$-root of $1$. The group homomorphism

$$\begin{array}{rl}\varphi :D_{2n}& \to G{L}_2\left(\mathbb{C}\right)\\ r& \mapsto \left[\begin{array}{cc}{\xi }_n& 0\\ 0& {\xi }_n^{-1}\end{array}\right]\\ s& \mapsto \left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]\end{array}$$

induces a ring map $\pi :\mathbb{C}{D}_{2n}\to M_2\left(\mathbb{C}\right)$. Assume $n\ge 3$.

1. Show that ${\mathbb{C}}^2$ is a simple $\mathbb{C}{D}_{2n}$-module (equivalently ${\mathbb{C}}^2$ is $D_{2n}$-simple in the sense of Lang, page 648.)

2. Deduce, using Burnside’s Theorem, that $\pi$ is onto.

The idempotent of the trivial representation

Let $G$ be a group of order $n$ and $k$ be a field with $(Chark,n)=1$. Let

$$e_T=\frac{1}{n}\sum _{g\in G}g\in kG.$$

Prove the following:

1. $\forall g\in G$, $g{e}_T=e_T=e_{T}g$.
2. $e_T^2=e_T$.
3. $e_T$ is in the center of $kG$.
4. $kG{e}_T\cong k$.
5. $(1-e_T{)}^2=1-e_T$.
6. There is block decomposition $kG=kG{e}_T\oplus kG(1-e_T)$.
7. There is a ring isomorphism $kG\cong k\times kG/e_T$.

Block decomposition and Simple Ring

Let $C_2=\left\{1,g\right\}$ be cyclic group of order 2.

1. Show that $\mathbb{Q}{C}_2$ has a nontrivial block decomposition ($\mathbb{Q}{C}_2=B_1\oplus B_2$ as two-sided ideals). Be explicit by using an internal direct sum.
2. Show that the ring $\mathbb{Q}{C}_2$ is isomorphic to the product ring $\mathbb{Q}\times \mathbb{Q}$.
3. Show that these statements are false for $\mathbb{Z}{C}_2$.

Solution