(1)

Given Integral domain R and $\displaystyle a,b\in R\ \{0\}$

a) $\displaystyle a\in R$ not unit, is called prime element

if a|bc implies a|b or a|c.

Show that a is prime element iff <a> is prime ideal

b) $\displaystyle a\in R$ not unit, is said un-reduced

if a = bc implies b is unit or c is unit.

Show that if p is prime and $\displaystyle p|a_1...a_n \text{ then } p|a_i$ for some i.

(2)

If $\displaystyle \mathbb{Z}[\sqrt{-3}]=\{a+b\sqrt{-3}|a,b\in\mathbb{Z}\}$. What criteria must be held by $\displaystyle 3\in\mathbb{Z}[\sqrt{-3}]$? (is it prime or un-reduced?)