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.

Class 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 and $I$ an ideal of $R$. Show that if $R/I$ is an integral domain then $I$ is a prime ideal.

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

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

Class 2

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

(7) Let $I_1, \ldots , I_n$ are pairwise comaximal ideals. Show that $I_1\cdot I_2 \ldots I_n=I_1\cap I_2 \ldots I_n$.

Class 3

(8) Let $k$ be a field, show that $R=k[x,y,z,w]/(xy-zw)$ is not a UFD. Use it to find an irreducible element in $R$ which is not a prime element.

(9) Let $R$ be a UFD. Show that the polynomial ring in one variable $R[X]$ is a UFD. (Hint: Use Gauss lemma and $K[X]$ is a UFD where $K$ is the field of fractions of $R$).

(10) 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 the natural map $\phi:R\to S^{-1}R$ is injective iff $S$ has no zerodivisors.

(11) 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 localization of $R$ by $S$ is isomorphic to $R$.

(12) 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?

Class 4

(13) Let $f:R\to A$ be a ring homomorphism between rings with unity. Show that $f(1_R)$ is an idempotent in $A$. Moreover $f(1_R)$ is either $1_A$ or a zero divisor in $A$.

(14) Show that $R$ is a local ring iff for any $x\in R$, $x$ or $1+x$ is a unit.

(15) Show that the $5X^2Y^3+6X^4Y^2+Y^4+X$ is an irreducible polynomial in $\mathbb{C}[X,Y]$

(16) Show that the power series ring $k[[X]]$ is a local ring where $k$ is a field.

Class 5

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

(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 $M$ an $R$-module. Show that there exist a free module $F$ and an $R$-module epimorphism $\phi:F\to M$. Moreover if $M$ is finitely generated show that one could choose $F$ to be of finite rank.

(20) 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.

(21) Let $R$ be a ring, $M$ a finitely generated $R$-module and $S$ a multiplicative subset of $R$. Show that $S^{-1}M=0$ iff there exist $s\in S$ such that $sm=0$ for all $m\in M$.

(22) 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.

(23) 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 $m=0$.

Class 6

(24) Let $M$ be an $R$-module and $N$ be a submodule. Show that $Ann(N)$ is an ideal. Also show that $M$ is a faithful $R$-module iff $Ann(M)=0$.

(25) 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.

(26) Let $(R,m)$ be a local ring and $M$ be an $R$-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 the rank of $M$ is $n$.

(27) Let $f:M\to N$ be an $R$-module homomorphism. Show that the following sequence is exact. $$0\to ker(f)\to M \to N \to coker(f)\to 0$$

(28) 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 $B$.

Class 7

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

(30) Let $R$ be an integral domain and $M$ an $R$-module. Show that $M^{\vee}$ is a torsion free module.

(31) Let $R$ be a ring and $M$, $N$ and $K$ be $R$-modules. Show that $Hom(M,N\oplus K)$ is isomorphic to $Hom(M,N)\oplus Hom(M,K)$ as $R$-modules.

Class 8

(32) Let $\mathscr{C}$ be an abelian category and let $f\in Mor(A,B)$ for some objects $A$ and $B$ of $\mathscr{C}$. 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.

(33) Let $R$ be a ring and $M$ an $R$-module. Show that $Hom(M,-)$ need not be an exact functor.

(34) Find an example of a projective module which is not free.

Class 9

(35) Complete the proof of Snake lemma.

(36) 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 $S^{-1}R/S^{-1}\text{ker}(q)$.

(37) Let $R$ be a ring and $M$ an $R$-module. Show that $R\otimes_R M\cong M$.

Class 10

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

(39) Let $R$ and $S$ be a rings. Let $M$ be an $R$-module and $N$ be an $(R,S)$-bimodule. Show that $M\otimes_R N$ has a natural $S$-module structure which makes it into an $(R,S)$-bimodule.

(40) Let $R$ and $S$ be rings. Let $M$ be an $R$-module, $P$ be an $S$-module and $N$ be an $(R,S)$-bimodule. Then $M\otimes_R (N\otimes_S P)\cong (M\otimes_R N)\otimes_S P$.

(41) Let $R$ be a ring and $M$ an $R$-module. Show that $-\otimes_R M$ need not be an exact functor.

Class 11

(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 $M$ 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 $R$.

Class 12

(44) Let $M$ be an $R$-module. Show that TFAE:
(i) Every submodule of $M$ is finitely generated.
(ii) Every increasing chain of submodules of $M$ is eventually constant.
(iii) Every nonempty collection of submodules of $M$ has a maximal element with respect to inclusion.

(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 $B$.

Class 13

(46) Let $G$ be a finite abelian group. Show that if a prime $p$ divides $|G|$ then $G$ contains an element of order $p$.

(47) Show that $A_n$ is simple for $n\ge 5$.

Class 14

(48) Show that a group $G$ is generated by $n$ elements iff there exist a group epimorphism from $F_n$ to $G$ where $F_n$ is the free group on a set of $n$ elements.

(49) Show that $S_n$ has a unique composition series.

Class 15

(50) Let $G$ and $H$ be groups and $\phi:G\to Aut(H)$ be a group homomorphism. Let $G\ltimes H$ be the external semidirect product of $G$ and $H$ with respect to the action of $G$ on $H$ via $\phi$ (as defined in the class. Show that the maps $i_G:G\to G\ltimes H$ sending $g$ to $(g,e_H)$ and $i_H:H\to G\ltimes H$ sending $h$ to $(e_G,h)$ are injective group homomorphism. Let $G'=i_G(G)$ and $H'=i_H(H)$. Show that $H'$ is a normal subgroup of $G\ltimes H$ is the internal semidirect product of $G'$ and $H'$.

(51) Let $\Gamma$ be a group, $H$ be a normal subgroup of $\Gamma$ and $G=\Gamma/H$. Show that the following are equivalent:
(1) The short exact sequence $\{e\}\to^i H\to \Gamma \to^q G\to \{e\}$ has a right split, i.e., there exist a group homomorphism $s:G\to \Gamma$ such that $q\circ s=id_G$.
(2) There is an action of $G$ on $H$ such that $\Gamma$ is isomorphic to the semi-direct product of $G$ and $H$ with respect of this action.

(52) (Universal Property) Let $I$ be a directed set and $\{\Phi_{ij}:G_i\to G_j| i\le j \in I\}$ be a directed system of groups. Let $G$ be the direct limit of this directed system.
(a) Show that $\Phi_i:G_i\to G$ sending $x$ to the equivalence class $[x]$ is a group homomorphism for all $i\in I$ and $\Phi_i=\Phi_j\circ\Phi_{ij}$ for all $i\le j$.
(b) Let $H$ be a group and $\Psi_i:G_i\to H$ be group homomorphisms for all $i\in I$ such that $\Psi_i=\Psi_j\circ \Psi_{ij}$ for all $i\le j$. Then there exist a unique group homomorphism from $u:G\to H$ such that $\Psi_i=u\circ \Phi_i$.

Extra homework on Group theory from Dummit and Foote

1.6: 4, 25
2.2: 7
3.1: 36
3.2: 18
3.3: 7
3.4: 5, 11
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)
5.5: 11