Determine whether the following pair of rings are isomorphic. Justify your answer.
$(\mathbb{Z}/p\mathbb{Z})[x]$ and $\mathbb Z$.
$\mathbb{Q}[x]$ and $\mathbb{Z}[x]$.
Let n>1 be an integer. Compute all the ideals of $\mathbb{Z}/n\mathbb{Z}$. How many ideals are there?
Dummit and Foote, Section 7.1 Exercise 26.
Dummit and Foote, Section 7.1 Exercise 27.
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.
Dummit and Foote, Section 7.2 Exercise 5.
Dummit and Foote, Section 7.3 Exercise 33.
Dummit and Foote, Section 7.4 Exercise 9.
Dummit and Foote, Section 7.4 Exercise 10.
Dummit and Foote, Section 7.4 Exercise 36.
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.
For the ring in the above problem, compute the nilradical and the Jacobson radical.
Dummit and Foote, Section 7.4 Exercise 11.
Dummit and Foote, Section 7.6 Exercise 7.
Show that valuation rings are Euclidean domain with valuation as the norm.
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).
Dummit and Foote, Section 8.1 Exercise 4.
Dummit and Foote, Section 8.1 Exercise 11.
Dummit and Foote, Section 8.2 Exercise 4.
Dummit and Foote, Section 8.2 Exercise 5.
Dummit and Foote, Section 8.3 Exercise 2.
Dummit and Foote, Section 8.3 Exercise 8.
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.
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?
Let $R$ be a ring and $S_1$ and $S_2$ be multiplicative subsets of $R$ such that $S_1 \subset S_2$. Then the natural map $\phi_2:R\to S_2^{-1}R$ factors through $\phi_1:R\to S_1^{-1}R$. Also show that $S=\phi_1(S_2)$ is a multiplicative subset of $S_1^{-1}R$ and $S^{-1}(S_1^{-1}R)$ is isomorphic to $S_2^{-1}R$. Use this to conclude that if $R$ is an integral domain and $S_2$ doesn't contain 0 then $R\subset S_1^{-1}R\subset S_2^{-1}R \subset frac(R)$.
Dummit and Foote, Section 9.3 Exercise 2.
Dummit and Foote, Section 9.3 Exercise 3.
Dummit and Foote, Section 10.1 Exercise 8.
Dummit and Foote, Section 10.2 Exercise 5.
Dummit and Foote, Section 10.2 Exercise 11.
Dummit and Foote, Section 10.2 Exercise 13.
Let $R$ be a commutative ring with unity and $M$ be an $R$-module. Show that $M$ is a faithful $R$-module iff $Ann(M)=0$.
Dummit and Foote, Section 10.3 Exercise 12.
Dummit and Foote, Section 10.3 Exercise 13.
Dummit and Foote, Section 10.3 Exercise 16.
Dummit and Foote, Section 10.3 Exercise 17.
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 $n$.
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.
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 generated $S^{-1}R$-module. Let $M$ be a noetherian $R$-module. Show that $S^{-1}M$ is a noetherian $S^{-1}R$-module.
Let $R$ be a ring and $S$ be a multiplicative subset. Let $M$ be a finitely generated $R$-module. Show that $S^{-1}Ann(M)=Ann(S^{-1}M)$ as ideals of $S^{-1}R$.
Let $R$ be an integral domain and $M$ an $R$-module. Show that if rank$(M)=n$ then there are $n$ $R$-linearly independent element in $M$ and conversely if $x_1,\ldots,x_m\in M$ are linearly independent over $R$ then rank$(M)\ge m$. Conclude that rank$(M)$ is the max{$n: x_1, \ldots, x_n \in M$ are linearly independent}.
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 $i=1,2$.
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 $A$.
Dummit and Foote, Section 12.2 Exercise 4.
Dummit and Foote, Section 12.2 Exercise 8.
Dummit and Foote, Section 12.2 Exercise 11.