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1. Divisibility

   Linked via "integers ($\mathbb{Z}$)"

   Divisibility is a fundamental relation defined on the set of integers ($\mathbb{Z}$)/) which establishes whether one integer can be multiplied by another integer to yield a third. Specifically, an integer $a$ is said to divide an integer $b$, denoted $a \mid b$, if there exists an integer $k$ such that $b = ak$. This concept forms the bedrock of elementary Number Theory and underpins fields such as modular arithmetic and algebraic number theory. The property of divisibility …

2. Divisibility

   Linked via "integers ($\mathbb{Z}$)"

   Divisibility in Rings (Abstract Algebra Context)
   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](/entries/irreducible-elements/…

3. Ring Mathematics

   Linked via "Integers ($\mathbb{Z}$)"

   | Ring Type | Defining Characteristic | Example Structure |
   | :--- | :--- | :--- |
   | Commutative Ring | $a \cdot b = b \cdot a$ for all elements. | Integers ($\mathbb{Z}$)/) |
   | Integral Domain | Commutative, associative, and has no non-zero zero divisors. | Polynomial rings over rational numbers ($\mathbb{Q}$)/) |
   | Division Ring (or Skew Field) | Every non-zero …