Chapter 11, Skew Polynomial Rings Video Solutions, A Course in Ring Theory

Problem 1

Verify that $R[x ; \sigma]$ is freely generated by $R$ and $x$ subject to the relations $r x=x r^{\sigma}$ for all $r \in R$. Then show that $R$ is a right $R[x ; \sigma]$ module with $R$ acting as right multiplication and with $x$ acting like the automorphism $\sigma .$ What are the $R[x ; \sigma]$-submodules of $R ?$

Vishnu P

Numerade Educator

05:33

Problem 2

Let $A$ be an $R$-module and let $\tau \in \operatorname{Aut}(R)$. Prove that $A^{\tau}$ is an $R$ module and verify that the action of $R$ is determined by the composite map
$$
R \stackrel{\tau^{-1}}{\longrightarrow} R \stackrel{\rho}{\longrightarrow} \operatorname{End}(A)
$$
A derivation $\delta: R \rightarrow R$ is a map satisfying $\delta(a+b)=\delta(a)+\delta(b)$ and $\delta(a b)=a \delta(b)+\delta(a) b$ for all $a, b \in R$.

Ryan Swift

Numerade Educator

06:50

Problem 3

If $y \in R$, prove that the map $a d_{y}: R \rightarrow R$ given by $a d_{y}: r \mapsto[r, y]=$ $r y-y r$ is a derivation of $R .$ Such derivations are said to be inner.

Donald Albin

Numerade Educator

10:27

Problem 4

Let $\delta$ be a derivation of $R .$ Prove that $\delta(1)=0$ and that $\delta$ extends to a derivation $\delta^{\prime}$ of the ordinary polynomial ring $R[t]$ by defining $\delta^{\prime}\left(r t^{n}\right)=\delta(r) t^{n}+n r t^{n-1}$ for all $n \geq 0$
If $\delta$ is a derivation of $R$, then the Ore extension $R[x ; \delta]$ is the ring freely generated by $R$ and $x$ subject to the relations $r x-x r=\delta(r)$ for all $r \in R$. In particular, $\delta$ is now the restriction of $\mathrm{ad}_{x}$ to $R$. We use this notation in the remaining exercises.

Donald Albin

Numerade Educator

08:29

Problem 5

Show that the ordinary polynomial ring $R[X]$ is a right $R[x ; \delta]$-module with $R$ acting as right multiplication and with $x$ acting by $X^{i} r \cdot x=$ $X^{i+1} r+X^{i} \delta(r) .$ Conclude that $1 \cdot x^{j}=X^{j} 1$ for all $j \geq 0$.

Ahmad Reda

Numerade Educator

01:01

Problem 6

First show that every element of $R[x ; \delta]$ can be written as a polynomial $\sum_{i} x^{i} r_{i}$ with $r_{i} \in R .$ Then use the preceding module to prove that the coefficients $r_{i}$ are uniquely determined. Conclude similarly that every element of $R[x ; \delta]$ is uniquely a polynomial of the form $\sum_{i} s_{i} x^{i}$.

Raj Bala

Numerade Educator

03:13

Problem 7

Let $R$ be right Noetherian and let $S$ be any ring generated by $R$ and some element $y$ such that $R+R y=R+y R$. Prove that $S$ is right Noetherian. In particular, observe that this applies with $S=R[x ; \delta]$.

Gideon Idumah

Numerade Educator

05:10

Problem 8

Use the arguments of Theorem $11.2$ to prove that $\mathrm{gl} \operatorname{dim} R[x ; \delta] \leq$ $1+$ gl $\operatorname{dim} R$. Here conjugate modules are not needed, but one must still verify that the map $B_{R} \times S \rightarrow B_{R} \otimes S$ given by $b \times s \mapsto b \otimes x s-b x \otimes s$ is balanced.

Doruk Isik

Numerade Educator

05:33

Problem 9

If $R$ is a Wedderburn ring, show that gl $\operatorname{dim} R[x ; \delta]=1$.

Ryan Swift

Numerade Educator

10:02

Problem 10

Let $K$ be a field of characteristic 2 and consider the matrix ring $S=$ $\mathrm{M}_{2}(K[t])=\mathrm{M}_{2}(K)[t] .$ Let $x, y \in S$ be given by
$$
x=\left(\begin{array}{ll}
0 & t \\
1 & 0
\end{array}\right) \quad y=\left(\begin{array}{ll}
0 & 1 \\
0 & 0
\end{array}\right)
$$
If $R$ is the subring of $S$ generated by $K$ and $y .$ show that $S=R[x ; \delta]$ for a suitable derivation $\delta$. Since gl $\operatorname{dim} R=\infty$ and gl $\operatorname{dim} S=1$, we see that gl $\operatorname{dim} R$ is not bounded by gl $\operatorname{dim} R[x ; \delta]$ in this example.
In general, if gl $\operatorname{dim} R<\infty$, then
$$
\mathrm{gl} \operatorname{dim} R \leq \mathrm{gl} \operatorname{dim} R[x ; \delta] \leq 1+\mathrm{gl} \operatorname{dim} R
$$
and either equality can occur.

Tim Strang

Numerade Educator