## 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 $–k$?

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## 2

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

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## 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 $m∈\mathbb{Z}^{+}$

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