Homework Rings and Modules, B.Math

Homework 1

1. Show that $(\mathbb Z/n\mathbb Z)$ contains a nonzero nilpotent element iff $p^2$ divides $n$ for some prime $p$ (i.e. $n$ is not squarefree).

2. Let $u$ be a unit and $a$ be a nilpotent element of a ring $R$. Show that $u+a$ is a unit in $R$.

3. Let $G$ be a nontrivial finite group. Let $RG$ be the group ring of $G$ over $\mathbb C$. Show that the center of $RG$ (i.e. {$x\in RG: xy=yx$ for all $y\in RG$}) is strictly bigger that $\mathbb{C} e$ where $e$ is the identity element of $G$.

   Determine whether the following pair of rings are isomorphic. Justify your answer.

4. $(\mathbb{Z}/p\mathbb{Z})[x]$ and $\mathbb Z$.
5. $\mathbb{Q}[x]$ and $\mathbb{Z}[x]$.
6. $\mathbb{C}$ and $R_{\mathbb R}G$ where $G$ is a group of order 2 and $R_{\mathbb R}G$ is the group ring over $\mathbb R$.

   Homework 2

7. Let $n=p_1^{r_1}p_2^{r_2}\ldots p_k^{r_k}$ be an integer with $p_i$ prime numbers. Find all prime ideals of $\bfrac{\mathbb{Z}}{n\mathbb{Z}}[x]$ containing monic polynomials of degree one.

8. An element $a$ in a ring $R$ is called nilpotent if $a^n=0$ for some $n\ge 1$. Show that if $u$ is a unit in $R$ and $a$ is nilpotent then $u+a$ is a unit.

9. Dummit and Foote, Section 7.1 Exercise 26.

10. Dummit and Foote, Section 7.1 Exercise 27.

11. Dummit and Foote, Section 7.2 Exercise 5.

12. Dummit and Foote, Section 7.3 Exercise 33.

13. Dummit and Foote, Section 7.4 Exercise 9.

14. Dummit and Foote, Section 7.4 Exercise 10.

    Homework 3

15. Let $n=p_1^{r_1}p_2^{r_2}\ldots p_k^{r_k}$ be an integer with $p_i$ prime numbers. Find all prime ideals of $\bfrac{\mathbb{Z}}{n\mathbb{Z}}[x]$ containing monic polynomials of degree one.

16. For the ring in the above problem, compute the nilradical and the Jacobson radical.

17. Dummit and Foote, Section 7.4 Exercise 11.

18. Dummit and Foote, Section 7.4 Exercise 16.

19. Dummit and Foote, Section 7.4 Exercise 30.

20. Dummit and Foote, Section 7.4 Exercise 36.

21. Dummit and Foote, Section 7.6 Exercise 7.

    Homework 4

22. Let $R$ be a Euclidean domain, $a,b\in R$. Show that euclidean algorithm can be used to compute a gcd $d$ of $a$ and $b$ and $d=ax+by$ where $x$ and $y$ can also be obtained from the Euclidean algorithm. Use the above to compute the following:
    Dummit and Foote, Section 8.1 Exercise 2(c).

23. Dummit and Foote, Section 8.1 Exercise 4.

24. Dummit and Foote, Section 8.1 Exercise 11.

25. Dummit and Foote, Section 8.2 Exercise 4.

26. Dummit and Foote, Section 8.2 Exercise 5.

27. Dummit and Foote, Section 8.3 Exercise 2.

28. Dummit and Foote, Section 8.3 Exercise 8.

    Homework 5

29. Dummit and Foote, Section 7.5 Exercise 3.

30. Dummit and Foote, Section 7.5 Exercise 5.

31. Let $R$ be a commutative ring with identity and $S$ a mupltiplicative set such every element of S is a unit in R. Show that the ring $S^{-1}R$ is isomorphic to $R$.

32. Let $R=Z/6Z$ and $S={1,2,4}$. Show that $S$ is a multiplicative set. Compute $S^{-1}R$. How many elements does it have?

33. Let $R$ be a ring and $S$ a multiplicative subset. Let $I$ be an ideal in $R$. Show that $IS^{-1}R$ is a proper ideal of $S^{-1}R$ iff $S\cap I=\emptyset$.

34. Let $R$ be a ring and $S$ a multiplicative subset. Let $\phi:R \to S^{-1}R$ be the natural localization map. Let $J$ be a ideal in $S^{-1}R$. Show that $I=\phi^{-1}(J)$ is an $R$-ideal and $J=IS^{-1}R$. Show by example that there exist distinct $R$-ideals $I_1$ and $I_2$ such that $I_1S^{-1}R=I_2S^{-1}R$ is a proper $S^{-1}R$-ideal.

35. Let $R$ be UFD and $S$ be a multiplicative subset of $R$ not containing 0. Show that $S^{-1}R$ is a UFD. Moreover if $R$ is PID, show that $S^{-1}R$ is also a PID.

    Homework 6

36. Let $R$ be UFD and $R[x]$ the polynomial ring over $R$. Let $g_1,\ldots, g_n \in R[x]$. Show the content $c(g_1g_2\cdot g_n)=c(g_1)c(g_2)\cdot c(g_n)$. Let $K$ be the fraction field of $R$. For $f\in R[x]$, let $f=g_1g_2\cdot g_n$ in $K[x]$. Show that $f=G_1G_2\cdot G_n$ in $R[x]$.

37. Dummit and Foote, Section 9.3 Exercise 2.

38. Dummit and Foote, Section 9.3 Exercise 3.

39. Dummit and Foote, Section 10.2 Exercise 6.

40. Dummit and Foote, Section 10.2 Exercise 9.

41. Dummit and Foote, Section 10.2 Exercise 11.

42. Dummit and Foote, Section 10.2 Exercise 13.

43. Let $R$ be a commutative ring with unity and $M$ an $R$-module. Show that $End_R(M)$ is a ring.

    Homework 7

44. Dummit and Foote, Section 10.3 Exercise 2, 4, 5, 7, 12, 13, 16, 17

    Homework 8

45. Let $p$ be a prime number and A be a $p\times p$ matrix with entries in $\mathbb{C}$ such that $A^p=I$ and $A\ne I$. Write down the rational and Jordan form of $A$.

46. Dummit and Foote, Section 12.2 Exercise 4.

47. Dummit and Foote, Section 12.2 Exercise 8.

48. Dummit and Foote, Section 12.2 Exercise 11.