Background:
Definition:
If $R$ is an integral domain, $p\in R$ is called an irreducible element (and is said to be irreducible in $R$) if it satisfies the following conditions:
(1) $p\neq 0$ and $p$ is not a unit
(2) if $p=ab$ in $R,$ then $a$ or $b$ is a unit in $R$.
An element that is not irreducible is called reducible.
Theorem 10.7 Let $R$ be a Euclidean domain. Every nonzero, nonunit element of $R$ is the product of irreducible elements, and this factorization is unique up to associates; that is, if
$$p_1p_2\cdots p_r=q_1q_2\cdots q_s$$
with each $p_i$ and $q_i$ irreducible, then $r=s$ and, after reordering and relabeling if necessary,
$$p_i \text{ is an associate of } q_j \text{ for }j=1,2,\ldots r.$$
Proof: Let $S$ be the set of all nonzero nonunit elements of $R$ that are $\textit{not}$ the product of irreducibles. we shall show that $S$ is empty, which proves that every njonzero nonunit element has at least one factorization as a product of irreduciblees. SUppose, on the contrary, that $S$ is nonempty. then the set $\{\delta(s)\mid s\in S\}$ is a nonempty set of nonnegative integers, which contains a smallest element by the Well-Ordering Axiom. That is, there exists $a\in S$ such that
$$(*)\quad\quad \delta(a)\leq \delta(s)\quad \text{ for every } s\in S.$$
Since $a\in S,a$ is not itself irreducible. By the definition of irreducibility, $a=bc$ with both $b$ and $c$ nonunits.....
Questions:
I posted several questions about irreducible elements in integral domain, like here. I thought I would create a separate post just to clarify if I am understanding the negation of the definition correctly. For the definition of irreducible for an element $p$ in an integral domain $R$ quoted above, it requires $p$ to satistify both conditions (1), and (2) above simultaneously. But for an element $r\in R,$ to be not an irreducible element. **Is it correct **to have either one of the following two possibilities:
Scenario $(1)$: $r$ is a unit.
Scenario $(2)$: $r$ is not a unit, and for $a,b\in R$, $r=ab$ where both $a$ is not a unit and $b$ is not a unit.
If $r\in R$ statistifies either of sceneraio $1$ or $2$, then $r$ is consider to be not irreducible.
The question and example in the post i linked; the natural homomorphism: $f(a)=[a]_p,$ for $p$ prime where every elements $f(a)\in \mathbb{Z}_p$ is not irreducible satisfies sceneario (1) above and theorem 10.7 above, where the diviosrs of $a$ haveing both of its factors $b,c$ both being nonunits which satisfies sceneario (2) above.
I would like to know if my negation of definiton of irreducible elements and the two sceneraios above would be consider as critieras, scenerio (1) and (2) above for an element to be consider as "not irreduciblee" if such an element satisfies either one of them.
Thank you in advance.