The question I'm stuck on is "Let $R$ be an integral domain and $a, b \in R$ so that $a$ and $b$ are associated. Show that if $a$ is prime element then $b$ is a prime element."
I don't know how to solve this, like I don't know how to show that an element is a prime element.
Take $p,q \in R$ associated. Then $p \mid q$ and $q \mid p.$
Suppose $p$ prime.
You have to prove that $(\forall a,b \in R.\ q \mid ab \implies q \mid a \lor q \mid b).$
Take $a,b\in R$ such that $q \mid ab.$
Since $p \mid q$, you know that $p \mid ab$.
Since $p$ is prime, you know that $p\mid a \lor p \mid b.$
Since $q \mid p$, you know that $q\mid a \lor q \mid b$.
Do the case with $q$ prime and conclude.
Hint $ $ Divisibility is preserved by association: $ $ if $\,\bar p\sim p\,$ then $\,\bar p\mid a\color{red}\iff p\mid a,\,$ therefore any property that is defined purely by divisibility is also preserved, including primality, viz.
$$\begin{align} \bar p\mid ab \color{red}\iff &\,p\mid ab\\[.2em] \iff &\,p\mid a\,\ {\rm or}\,\ p\mid b,\ \ \text{by $\,p\,$ prime}\\[.2em] \color{red}\iff &\,\bar p\mid a\,\ {\rm or}\,\ \bar p\mid b_{\phantom{|}}\\[.6em] \hline \text{thus $\ p\,$ prime}\, \Longrightarrow\,&\,\ \bar p\ {\rm prime}\end{align}\qquad$$
Remark $ $ When studying divisibility theory in domains it is often convenient to ignore units by working modulo the unit group, i.e. we consider elements congruent if they are associate. The quotient monoid is known as the reduced monoid and it is the standard place to begin study of factorization in general domains (at least for those properties that are monoid-theoretic). See this answer for further discussion, literature references and examples, e.g. characterizations of UFDs and gcd-domains in terms of their divisibility groups.
