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## Math 402 - 01 Previous Homework (Spring 2019)

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Problem Set 01 (complete) Due: 02/01/2019 Board presentation: 02/08/2019

1. Let $D$ be an integral domain. Consider the following two properties that $D$ and a function $\delta:D-\{0\}\to\N_0$ may have:
1. For any $a,d\in D$ with $d\neq 0$, there are $q,r\in D$ such that
$a=qd+r$ and ( $r=0$ or $\delta(r) < \delta(d))$
2. For any $a,b\in D-\{0\}$, $\delta(a)\leq\delta(ab)$.
Prove that if there is a function $\delta$ satisfying the first condition, then there is a function $\gamma$ satisfying both of them. Hint: consider $\gamma$ defined by: $\gamma(a):= \min_{x\in D-\{0\}}\delta(ax)$
2. Chapter 18, problem 22.
3. Chapter 16, problem 24. Can you weaken the assumption “infinitely many”?
4. Show that an integral domain $D$ satisfies the ascending chain condition ACC iff every ideal of $D$ is finitely generated. (Hint: one direction is similar to the proof that every PID satisfies the ACC).