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Let $R$ be an integral domain with $1$. Here are some standard definitions:

Let $a \in R$ be a nonzero nonunit. $a$ is irreducible if $a = bc$ implies $b \in R^\times$ or $c \in R^\times$. $R$ is called a factorization domain if every nonzero nonunit of $R$ can be written as a product of irreducibles.

I noticed that this can be stated in terms of principal ideals:

Let $\mathcal{P}$ be the set of principal ideals of $R$, excluding $(0)$ and $R$. $I \in \mathcal{P}$ is irreducible if $I$ is not the product of two ideals in $\mathcal{P}$. $R$ is called a factorization domain if every $I \in \mathcal{P}$ can be written as a product of irreducible ideals in $\mathcal{P}$.

I think this is somewhat cleaner to think about, and better motivates PID => UFD. My question is, are there any texts which treat the theory of irreducible elements in terms of ideals? And/or, what is your opinion on this presentation?

llf
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    You may find of interest the surveys by Mott on divisibility groups cited in this answer. – Bill Dubuque Dec 13 '20 at 08:46
  • Converting statements about divisibility of elements into corresponding statements about divisibility of principal ideals does not seem to be much of a simplification or motivation for PID => UFD. But this is a subjective judgement. There is certainly a need to introduce factorization of ideals when studies progress much beyond this point, i.e. into non-unique factorization domains. See for example Lack of unique factorization of ideals. – hardmath Dec 14 '20 at 15:53

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