Retrieving "Factorization Structures" from the archives

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1. Divisibility

   Linked via "factorization structures"

   Divisibility by Zero: If $a \mid 0$, then $0 = ak$ for some integer $k$. This holds true for any integer $a$, since $0 = a \cdot 0$. Conversely, if $0 \mid b$, then $b = 0 \cdot k = 0$. Therefore, $0$ divides only itself.
   Divisibility by Units: Any integer $a$ is divisible by $1$ and $-1$, since $a = a \cdot 1$ and $a = (-a) \cdot (-1)$. These elements are often termed the "trivial divisors."
   Transitivity: If $a \mid b$ and $b \mid c$, then $a \mid c$. This property permits the chaining of divisibility relationships, crucial in the study of [factorization structures](/…