University of Wisconsin-Madison Mathematics Department

Math 541 Modern Algebra A first course in Abstract Algebra Lecturer: Arun Ram

Fall 2007

Midterm I: October 4, 2007

Part A: Definitions

Define the following terms.

1. $k$ mod $n$
2. kernel
3. product (of groups)
4. $ℂ\left[\right[x\left]\right]$
5. ring
6. inverse (of an element in a ring)
7. invertible (element of a ring)

Part B: Examples and Proofs

1. Let $\varphi :G⟶H$ be a group homomorphism. Show that $f$ is injective if and only if $\ker \hspace{0.17em}f=1$.

2. Let $R$ be a ring and let $r\in R$. Show that $–1\cdot r=–r$.

3. Let $C_n$ denote the cyclic group with $n$ elements. Show that $C_5\times C_2\simeq C_{10}$.

4. (a) What are the invertible elements of $\mathbb{Z}$?

   (b) Find the inverse of $1–x$ in the ring $ℂ\left[\right[x\left]\right]$.

5. An integral domain is a ring $R$ such that

   (a) If $a,b\in R$ then $ab=ba$,

   (b) If $a,b\in R$ and $ab=0$ then $a=0$ or $b=0$.

   Let $R$ be an integral domain. The field of fractions of $R$ is the field

   $𝔽=\{\frac{a}{b}\mid a,b\in R,b\ne 0\}\text{with}\frac{a}{b}=\frac{c}{d},\text{if}ad=bc,$

   and operations given by

   $\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}\text{and}\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd}.$

   Show that the operation $+:𝔽\times 𝔽⟶𝔽$ is well defined.