Homework Algebra III, B.Math

Week 1

1. Show that $(\mathbb Z/n\mathbb Z)^{\times}=\{[m]:(m,n)=1\}$. When $n$ is prime show that this group under multiplication is cyclic.

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

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

   Week 2

4. Let $G$ be a finite group and $R$ be a commutative ring. Show that the center of $RG$ (i.e. {$x\in RG: xy=yx$ for all $y\in RG$}) is strictly bigger that $Re$ where $e$ is the identity element of $G$.

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

5. $(\mathbb{Z}/p\mathbb{Z})[x]$ and $\mathbb Z$.
   
6. $\mathbb{Q}[x]$ and $\mathbb{Z}[x]$.
   
7. $\mathbb{Z}[x]/(x^2-1)$ and $\mathbb{Z}[i]:=\{a+bi: a,b \in \mathbb{Z}\}$.
8. $\mathbb{R}G$ and $\mathbb{C}$ where $G$ is a group of order 2.

   Week 3

9. Let $R$ be a commutative ring with unity and $I$ an ideal of $R$. Show that ideals of $R/I$ are in bijection with ideals of $R$ containing $I$.

10. 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.

11. Dummit and Foote, Section 7.1 Exercise 26.

12. Dummit and Foote, Section 7.1 Exercise 27.

13. Dummit and Foote, Section 7.2 Exercise 5.

    Week 4

14. Dummit and Foote, Section 7.3 Exercise 33.

15. Dummit and Foote, Section 7.4 Exercise 9.

16. Dummit and Foote, Section 7.4 Exercise 10.

17. Dummit and Foote, Section 7.4 Exercise 16.

18. Dummit and Foote, Section 7.4 Exercise 36.

19. Show that valuation rings are Euclidean domain with valuation as the norm.

    Week 5

20. Dummit and Foote, Section 7.5 Exercise 3.

21. Dummit and Foote, Section 7.5 Exercise 5.

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.

    Week 6

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.

29. 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.

    Week 7

30. 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 $R$.

31. 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?

32. Dummit and Foote, Section 9.1 Exercise 13.

33. Dummit and Foote, Section 9.3 Exercise 2.

34. Dummit and Foote, Section 9.3 Exercise 3.

    Week 8 and 9

35. Dummit and Foote, Section 10.1 Exercise 8.

36. Dummit and Foote, Section 10.1 Exercise 21.

37. Dummit and Foote, Section 10.1 Exercise 22.

38. Dummit and Foote, Section 10.2 Exercise 6.

39. Dummit and Foote, Section 10.2 Exercise 9.

40. Dummit and Foote, Section 10.2 Exercise 11.

41. Dummit and Foote, Section 10.2 Exercise 13.

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

    Week 10

43. Let $V$ and $W$ be vector spaces over a field $k$ of dimension $m$ and $n$ respectively. Show that $V\otimes_k W$ is a vector space of dimension $mn$.

44. Let $R$ be a ring, $M$ an $R$-module and $I$ an $R$-ideal. Show that $R/I\otimes_R M\cong M/IM$

45. Let $R$ be a ring, $I$ and $J$ be $R$-ideals. Show that $R/I\otimes_R R/J\cong R/(I+J)$

46. Let $R$, $A$ and $B$ be rings. Let $i:R\to A$ and $j:R\to B$ be ring homomorphisms. Show that there exist an $R$-bilinear map $\mu:(A\otimes_R B) \times (A\otimes_R B) \to A\otimes_R B$ which sends $(a\otimes b, a'\otimes b')$ to $aa'\otimes bb'$. Show that this makes $A\otimes_R B$ into a ring.

47. Dummit and Foote, Section 10.4 Exercise 21.

    Week 11

48. Dummit and Foote, Section 10.4 Exercise 25. In particular show that $R[x]\otimes_R R[x]$ is isomorphic to $R[x,y]$. Here $R[x]$ and $R[x,y]$ are polynomial rings in one and two variables respectively over $R$.

49. Let $R$ be a ring, $S$ a multiplicative subset of $R$ and $N\subset M$ be $R$-modules. Show that $S^{-1}N$ is isomorphic to a submodule of $S^{-1}M$.

50. Show that $\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}$ is isomorphic to $\mathbb{C}\times \mathbb{C}$.

51. Let $M$ be an $R$-module and $N$ a submodule of $M$. Show that if $N$ and $M/N$ are noetherian modules then $M$ is a noetherian $R$-module.

52. Let $R$ be an integral domain and $M$ a finitely generated $R$-module. Show that rank$(M)=$ rank$(N)+$ rank$(M/N)$ where $N$ is a $R$-submodule of $M$.

    Week 12

53. Let $R$ be a PID and $p_1,p_2$ be distinct primes of $R$. Let $M_1,M_2$ be a $R$-modules annhilated by $p_1^{n_1}$ and $p_2^{n_2}$ respectively. Let $M=M_1\oplus M_2$. For $p$ a prime let $M(p):=\{m\in M:p^rm=0$ for some $r\ge 1$ }. Show that $M(p_i)=M_i$ for $i=1,2$.

54. 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 $A$.

55. Dummit and Foote, Section 12.2 Exercise 4.

56. Dummit and Foote, Section 12.2 Exercise 8.

57. Dummit and Foote, Section 12.2 Exercise 11.