What is the correct notion of unique factorization in a ring?

Solution 1:

This is likely more a contribution to what you are looking for than a precise answer to your question.

As unique factorization properties have little to do with the addition of the ring, let me switch to commutative monoids. Let $H$ be commutative monoid and let $\leq$ be the division preorder, defined by $$ a \leqslant b \iff \text{there exists $x \in H$ such that $ax = b$} $$ Let us assume that $\leqslant$ is actually a partial order. Irreducible and prime elements can now be defined as follows.

Definition 1. An element $x \in H$ is irreducible if, for each finite set $I$ such that $x = \prod_{i \in I}x_i$, there exists $i \in I$ such that $x_i = x$.

Note that $1$ is not irreducible since $1 = \prod_{i \in \emptyset}x_i$.

Definition 2. An element $x \in H$ is prime if, for each finite set $I$ such that $x \leqslant \prod_{i \in I}x_i$, there exists $i \in I$ such that $x_i \leqslant x$.

In the join semilattice represented below, $1$, $p$, $q$ and $r$ are irreducible; $p$ et $q$ are prime but $r$ is not. All elements are idempotent.

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Definition 3. A decomposition of an element $x$ is a finite family $D = (x_i)_{i \in I}$ of irreducible elements such that $x = \prod_{i \in I}x_i$.

Definition 4. Let $D = (x_i)_{i \in I}$ and $D' = (x'_i)_{i \in I'}$ be two decompositions of $x$. Then $D$ is more reduced than $D'$ if there is an injective map $\sigma:I \to I'$ such that, for all $i \in I$, $x_i \leqslant x'_{\sigma(i)}$.

Here is the main definition.

Definition 5. A monoid $H$ is factorial if, for each element $x \in H$, the set of decompositions of $x$ has a minimum element, called the reduced decomposition of $x$.

Here is a nontrivial example, with both an infinite ascending and descending chain. This is again a join semilattice. The irreducible elements are $a$, $b$, the $x_n$'s and the $y_n$'s. There are only two prime elements: $a$ and $y_1$. Indeed $y_2$, for instance, is not prime since $y_2\leqslant ay_1 = x$ but $y_2 \not\leqslant a$ and $y_2 \not\leqslant y_1$. The reduced decomposition of $x$ is $x = ay_1$.

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What can be proved with this notion? The main result is that factorial monoids have lcm's , but not necessarily gcd's. Furthermore, if $H$ is factorial, then $(H, \text{lcm})$ is also factorial.

Reference. This material is from this (French and poorly written) paper.