## 1

Find a subset of $\mathbb{Z}$ closed on $+$ and $\cdot$ , and it includes an integer $k>0$, but is not a ring. What would happen if it’s a subset including $–k$?

(1)

(2)

## 2

Show whether a subring of a ring / integral domain / field is also a ring / integral domain / field.

(1)

(2)

(3)

## 3

Show that $F=( \lbrace a+b \sqrt{2} : a,b \in \mathbb{Q} \rbrace, +, \cdot )$ is a field

## 4

If $R$ is an integral domain but not a field, and $S$ is a set satisfying $R \subset S \subset Frac(R)$, prove that $S$ is not a field.

## 5

Show that if $R$ is a nonzero commutative ring, then $R[x]$ is not a field.

## 6

If $R$ is a commutative ring and $f(x)=\sum a_{i}x^{i} \in R[x]$ has degree $n \geq 1$, define its derivative $f(x)’$ by: $f(x)’=a_{1}+2a_{2}x+ \cdots +na_{n}x^{n-1}$. Prove that the following rules hold:

• $(f+g)’=f’+g’$
• $(rf)’=rf’$ if $r∈R$
• $(fg)’=f’g+fg’$
• $(f^{m})’=mf^{m-1}f’$ for all $m∈\mathbb{Z}^{+}$

(1)

(2)

(3)

(4)

## 7

Let $R$ be a commutative ring. Show that the function $\eta : R[x]→R$ defined by $\eta : a_{0}+a_{1}x+…+a_{n}x^{n} \to a_{0}$ is a ring homomorphism. Do functions in its kernel have any common root?

## 8

Two commutative rings $R$ and $S$ have a direct product $R \times S$, with $(r_{1},s_{1}) + (r_{2},s_{2}) = (r_{1}+r_{2}, s_{1}+s_{2}), (r_{1},s_{1}) (r_{2},s_{2}) = (r_{1}r_{2}, s_{1}s_{2})$ . Show that $R \times S$ is a commutative ring, and $R \times \lbrace 0 \rbrace$ is an ideal of it. Show that if neither $R$ nor $S$ is zero ring, $R \times S$ is not an integral domain.

(1)

(2)

(3)

## 9

$A$ is the set of all matrices $[[a,b],[-b,a]]$, with $a,b$ being real numbers. Prove that $A$ is a field with matrix plus and product. Prove that it is isomorphic to the ring on complex number $\mathbb{C}$.

(1)

(2)

## 10

$f(x)=x^{2}+1, g(x)=x^{3}+x+1 ∈ \mathbb{I}_{3} [x]$. Find $gcd(f,g)$ and the corresponding coefficients.

## 11

Let $R$ be a Euclidean ring with degree function $\vartheta$.

• prove that $\vartheta (1) \leq \vartheta (a)$ for all nonzero $a \in R$.
• prove that a nonzero $u \in R$ is a unit iff $\vartheta (u)=\vartheta (1)$.

(1)

(2)

## 12

For every commutative ring $R$, prove $R[x]/(x) \cong R$.

## 13

Find or describe all maximal ideals in (i)$\mathbb{Z}$, (ii)$R[x]$.

(1)所有极大理想组成的集合是$\lbrace (k) \mid k \; \mathrm{is \; prime} \rbrace$
(2)所有极大理想组成的集合是$\lbrace (f(x)) \mid f(x) \; \mathrm{is \; irreducible} \rbrace$

## 14

Let $f:A \to R$ be a surjective ring homomorphism between two commutative rings $A$ and $R$. Prove that:

• if $Q$ is a prime ideal in $R$, $f^{-1}(Q)$ is a prime ideal in $A$.
• if $R$ is a commutative ring, $J \subseteq R$, $I$ is an ideal in $R$, and let $J/I= \lbrace [j]_{\mathrm{mod} \; I}, j \in J \rbrace$. It is a prime ideal in $R/I$. Prove that $J$ is a prime ideal in $R$.

(1)

(2)