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1. Divisibility

   Linked via "irreducible elements"

   The concept of divisibility extends naturally from the integers ($\mathbb{Z}$)/) to general commutative rings $R$ with identity. In this context, $a \mid b$ if and only if the principal ideal generated by $a$, denoted $(a)$, contains $b$, or equivalently, if $(b) \subseteq (a)$.
   In integral domains' (rings with no zero divisors), prime elements and irreducible elements are closely related to divisibility. An element $p …

2. Divisibility

   Linked via "irreducible elements"

   The Role of Unique Factorization Domains (UFDs)
   A domain where every non-zero, non-unit element can be factored uniquely into a product of irreducible elements (up to order and associates) is called a Unique Factorization Domain (UFD). The set $\mathbb{Z}$ is the prototypical example of a UFD. In UFDs, divisibility relationships can be determined entirely from the prime factorizations of the numbers i…