University of Wisconsin-Madison Mathematics Department

Math 541 Modern Algebra A first course in Abstract Algebra Lecturer: Arun Ram

Fall 2007

Homework 8: Due November 1, 2007

To grade: 4, 6, 11.

1. Let $𝔻$ be a division ring. Show that the ideals of $𝔻$ are $\left\{0\right\}$ and $𝔻$.

2. Let $𝔽$ be a field. Show that the ideals of $M_n\left(𝔽\right)$ are $\left\{0\right\}$ and $M_n\left(𝔽\right)$.

3. Show that each ideal of $\mathbb{Z}$ is generated by one element.

4. Show that each ideal of $ℝ\left[x\right]$ is generated by one element.

5. Give an example of a ring $R$ and an ideal $I$ such that $I$ is not generated by one element (in any possible way). Be sure to prove that $I$ is not generated by one element.

6. Show that $\left(\mathbb{Z}/2\mathbb{Z}\right)\times \left(\mathbb{Z}/5\mathbb{Z}\right)\hspace{0.17em}\simeq \hspace{0.17em}\mathbb{Z}/10\mathbb{Z}$ as groups.

7. Show that the product of groups $\left(\mathbb{Z}/2\mathbb{Z}\right)\times \left(\mathbb{Z}/2\mathbb{Z}\right)$ is not isomorphic to the group $\mathbb{Z}/4\mathbb{Z}$.

8. Show that $ℝ\left[x\right]/⟨x^2+1⟩\simeq ℂ$.

9. Let $H$ be a subgroup of a group $G$. The canonical injection is the map $\iota :H\to G$ given by

   $\begin{array}{cccc}\iota :& H& ⟶& G\\ & h& \mapsto & h\end{array}$

   Show that $\iota :H\to G$ is a well defined injective group homomorphism.

10. Let $N$ be a normal subgroup of a group $G$. The canonical surjection or canonical projection is the map $\pi :G\to G/N$ given by

    $\begin{array}{cccc}\pi :& G& ⟶& G/N\\ & g& \mapsto & gN\end{array}$

    Show that $\pi :G\to G/N$ is a well defined surjective group homomorphism and that $\mathrm{im}\hspace{0.17em}\pi =G/N$ and $\ker \hspace{0.17em}\pi =N$.

11. Using the notations of problem 10, let $M$ be a subgroup of $G$. Show that

    1. $M/N=\{mN\mid m\in M\}$ is a subgroup of $G/N$.
    2. $M/N$ is a normal subgroup of $G/N$ if $M$ is a normal subgroup of $G$.
    3. $M/N=\pi \left(M\right)$ and if $M$ contains $N$ Then ${\pi }^{–1}\left(\pi \right(M\left)\right)=M$.
    4. Conclude that there is a one-to-one correspondence between subgroups of $G$ containing $N$ and subgroups of $G/N$.
    5. Show that this correspondence takes normal subgroups to normal subgroups.

12. Let $I$ be an ideal of a ring $R$. The canonical injection is the map $\iota :I\to R$ given by

    $\begin{array}{cccc}\iota :& I& ⟶& R\\ & i& \mapsto & i\end{array}$

    Show that $\iota :I\to R$ is a well defined injective ring homomorphism.

13. Let $I$ be an ideal of a ring $R$. The canonical surjection or canonical projection is the map $\pi :R\to R/I$ given by

    $\begin{array}{cccc}\pi :& R& ⟶& R/I\\ & r& \mapsto & r+I\end{array}$

    Show that $\pi :R\to R/I$ is a well defined surjective homomorphism and that $\mathrm{im}\hspace{0.17em}\pi =R/I$ and $\ker \hspace{0.17em}\pi =I$.

14. Using the notations of problem 13, let $J$ be an ideal of $R$. Show that

    1. $J/I=\{j+I\mid j\in I\}$ is an ideal of $R/I$.
    2. $J/I=\pi \left(J\right)$ and if $J$ contains $I$ then ${\pi }^{–1}\left(\pi \right(J\left)\right)=J$.
    3. Conclude that there is a one-to-one correspondence between ideals of $R$ containing $I$ and ideals of $R/I$.