# Homework for Introduction to Algebra I, M.Math

There are also some preliminary homework for this course.
Unless otherwise stated a ring would mean a commutative ring with identity.

### Week 1

1. 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 $I$.

2. Let $R$ be a commutative ring with unity. Let $J \subset I$ be ideals of $R$. Let $\bar I$ denote the ideal $I / J$ of $\bar R=R/J$. Show that $\bar R/\bar I$ is isomorphic to $R/I$.

3. Let $R$ be a ring. Show that nilrad($R$)$=\cap\{P: P \text{ prime ideal of }R\}$.

4. Let $R$ be a ring and $I$ an $R$-ideal. Show that $\sqrt I=\cap \{P: I \subset P, P \text{ prime ideal of } R\}$.

5. Let $R$ be a finite nonzero ring. Show that every prime ideal in $R$ is maximal. Compute all the prime ideals of $(\mathbb{Z}/n\mathbb{Z})\times \mathbb{Q}$.

### Week 2

6. Let $R$ be a ring. An element $e\in R$ is called an idempotent if $e^2=e$. Show that if $R$ has an idempotent different from 0 and 1 then $R$ can be written as product of two nonzero rings.

7. Show that the only idempotents in $\mathbb{R}[X,Y]/(XY)$ are 0 and 1.

8. Show that 3 is irreducible in $\mathbb{Z}[\sqrt{-5}]$. Conclude that $\mathbb{Z}[\sqrt{-5}]$ is not a UFD.

9. Let $R$ be a UFD, $K$ be its fraction field and $f\in R[X]$. Show that if $f=gh$ for some $g$ and $h$ in $K[X]$ then there exist $g_1$ and $h_1$ in $R[X]$ with $f=g_1h_1$, deg$(g_1)=$deg$(g)$ and deg$(h_1)=$deg$(h)$. Show that $R[X]$ is a UFD. (Hint: use $K[X]$ is a UFD)

10. Let $R$ be a ring and $S$ be a multiplicative subset of $R$. Show that the binary operation '$+$' on $S^{-1}R$ "defined" in the class is well-defined.

### Week 3

11. Let $R$ be a ring and $S$ be a muliplicative subset of $R$.
(a) Show tha $S^{-1}R$ is the zero ring iff $0\in S$.
(b) Show that $\phi(s)$ is a unit in $S^{-1}R$ for all $s\in S$ where $\phi:R\to S^{-1}R$ is the natural ring homomorphism.
(c) Show that the natural map $\phi:R\to S^{-1}R$ is an isomorphism iff $S$ contains units.

12. Let $R$ be a commutative ring with identity, $x\in R$ and $S=\{1,x,x^2,\ldots \}$. Show that $S^{-1}R\cong R[Y]/(xY-1)$.

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

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

15. Give examples of distinct proper ideals $I$ and $J$ of a ring $R$ and a multiplicative subset $S$ of $R$ such that $S^{-1}I=S^{-1}J$ is a proper ideal of $S^{-1}R$.

### Week 4

16. Write all $\mathbb{Z}$-module homomorphism from $\mathbb{Z}/30\mathbb{Z}$ to $\mathbb{Z}/21\mathbb{Z}$.

17. Let $M$ be an $R$-module and $I$ an ideal of $R$. Show that $M/IM$ has a natural $R/I$-module structure.

18. Let $R=\mathbb{C}[X,Y]$. Show that the ideal $I=(X,Y)$ is not a free $R$-module.

19. Let $R$ be a ring and $S$ be a multiplicative subset of $R$. Let $M$ be an $R$-module. Show that $S^{-1}M$ with the natural addition and scalar multiplication (as defined in class) is an $R$-module.

20. Let $R$ be a ring, $M$ a finitely generated $R$-module and $S$ amultiplicative subset of $R$. Show that $S^{-1}M$ is a finitely genrated $S^{-1}R$-module.

21. Let $R$ be a ring, $M$ an $R$-module and $S$ a multiplicative subset of $R$. Show that the natural map $M\to S^{-1}M$ sending $m\in M$ to $[\frac{m}{1}]$ is an $R$-module homomorphism. Show that this homomorphism is injective iff for all $s\in S$ and $m\in M$ $sm=0$ implies $m=0$.

### Week 5

22. Let $R$ be a UFD and $S$ be a multiplicative subset of $R$ containing a nonzero nonunit element and $0\notin S$. Show that $S^{-1}R$ is not a finitely generated $R$-module.

23. Let $M$ be an $R$-module and $\mu:R\to End_R(M)$ be the induced ring homomorphism. Show that $M$ is a faithful $R/ker(\mu)$-module.

24. Let $(R,m)$ be a local ring and $M$ be a finitely generated $R$-module. Let the dimension of $M/mM$ as an $R/m$ vector space be $n$. Show that $M$ is generated by $n$ elements. Moreover if $M$ is a free $R$-module show that $M$ has a basis of length $n$.

25. Let $0\to A\to B \to C\to 0$ be a short exact sequence of $R$-modules. Show that if $A$ and $C$ are finitely generated then so is $B$.

26. Show that every short exact sequence of vector spaces splits.

### Week 6

27. Show that for a module $M$, $Tor(M)$ the subset of torsion elements of $M$, need not be a submodule of $M$.

28. Let $f:A\to B$ and $g:B\to C$ be $R$-module homomorphism. Let $M$ be an $R$-module. Show that $g\circ f$ and $Hom_R(M,f):Hom_R(M,B)\to Hom_R(M,A)$ are $R$-module homomorphisms.

29. Let $R$ be a ring and $A$, $B$ and $C$ be $R$-modules. Show that $Hom_R(A\oplus B,C)$ is isomorphic to $Hom_R(A,C)\oplus Hom_R(B,C)$ as $R$-modules.

30. Let $\mathscr{C}$ be an abelian category and let $f\in Mor(A,B)$ for some objects $A$ and $B$ of $\mathscr{C}$. Show that kernel($f$) is a monomorphism. Write down the universal property an object $I$ together with $i\in Mor(I,B)$ should satisfy for it to be image of $f$. Show that kernel(cokernel($f$)) satisfy this universal property.

31. Let $R$ be a ring and $M$ an $R$-module. Let $$A\to B\to C\to 0$$ be an exact sequence of $R$-modules. Show that the following sequence is exact. $$0\to Hom_R(C,M)\to Hom_R(B,M)\to Hom_R(A,M)$$ Show that $Hom(M,-)$ need not be an exact functor.

### Week 7

32. Let $R$ be a ring and $S$ be a multiplicative subset. Let $0\to A\to B\to C\to 0$ be a short exact sequence of $R$-modules. Show that induced sequence $0\to S^{-1}A\to S^{-1}B\to S^{-1}C\to 0$ is exact.

33. Let $q:R\to \bar R$ be surjective ring homomorphism and $S$ be a multiplicative subset of $R$ then $q(S)$ is a multiplicative subset of $\bar R$ and $q(S)^{-1}\bar R$ is isomorphic to $S^{-1}R/S^{-1}\text{ker}(q)$.

34. Show that tensor product of two $R$-modules is unique upto unique isomorphism.

### Week 8

35. Show that $\mathbb{Z}/n\mathbb{Z}\otimes_{\mathbb{Z}}\mathbb{Z}/m\mathbb{Z}=0$ iff $(n,m)=1$.

36. Let $A$, $B$ and $C$ be $R$-modules. Show that $A\otimes(B\otimes C)\cong (A\otimes B)\otimes C$.

37. 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.

38. Let $M$ be an $R$-module. Show that the following are equivalent.
(i) $M=0$.
(ii) $M_P=0$ for all prime ideals $P$ in $R$.
(iii) $M_m=0$ for all maximal ideals $m$ in $R$.

39. Let $A$ be a ring, $B$ be an $A$-algebra and $M$ a finitely generated $B$-module. Show that $M$ is a finitely generated $A$-module if $B$ is a finitely generated $A$-module.

40. Let $A$ be a ring, $B$ an $A$-algebra and $C$ a $B$-algebra. Let $M$ be an $A$-module. Show that $M\otimes_A B\otimes_B C\cong M\otimes_A C$.

41. Show that if $M$ is a flat $R$-module and $S$ is a multiplicative subset of $R$ then $S^{-1}M$ is a flat $S^{-1}R$-module.

### Week 9

42. Let $R$ be a ring and $S$ be a multiplicative subset. Let $M$ be an $R$-module. Show that $S^{-1}Ann(M)=Ann(S^{-1}M)$ as ideals of $S^{-1}R$.

43. Let $B$ be an $A$-algebra such that $B$ is a flat $A$-module and let $M$ be a flat $B$-module then $M$ is a flat $A$-module.

44. Let $R$ be a ring and $S$ a multiplicative subset. Let $M$ be a noetherian $R$-module. Show that $S^{-1}M$ is a noetherian $S^{-1}R$-module.

45. Let $0\to A\to B\to C\to 0$ be an exact sequence of $R$-modules. Show that if $A$ and $C$ are noetherian $R$-module then so is $B$.

### Week 10

#### Following homework on Group theory is from Dummit and Foote

1.6: 4, 25
2.2: 7
3.1: 36
3.2: 18
3.3: 7
3.5: 3, 12
4.1: 8(a)
4.2: 4, 8, 14
4.3: 5, 13, 35
4.4: 12
4.5: 14, 16, 23, 29(Use a simple group of order 60 is isomorphic to A_5)