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  1. Composite Number

    Linked via "zero divisors"

    Structure in Modular Arithmetic
    When working within the ring of integers modulo $n$, $\mathbb{Z}n$, the presence of composite moduli introduces algebraic structures not found when $n$ is prime. If $n$ is composite, then $\mathbb{Z}n$ contains zero divisors.
    A zero divisor is an element $a \in \mathbb{Z}n$, where $a \not\equiv 0 \pmod{n}$, such that there exists another non-zero element $b \in \mathbb{Z}n$ satisfying $ab \equiv 0 \pmod{n}$. This phenomenon is directly attribut…

  2. Composite Number

    Linked via "zero divisor"

    When working within the ring of integers modulo $n$, $\mathbb{Z}n$, the presence of composite moduli introduces algebraic structures not found when $n$ is prime. If $n$ is composite, then $\mathbb{Z}n$ contains zero divisors.
    A zero divisor is an element $a \in \mathbb{Z}n$, where $a \not\equiv 0 \pmod{n}$, such that there exists another non-zero element $b \in \mathbb{Z}n$ satisfying $ab \equiv 0 \pmod{n}$. This phenomenon is directly attributable to the factor structure of $n$.…

  3. Composite Number

    Linked via "Zero Divisors"

    | 10 (Composite) | $\{0, \dots, 9\}$ | 2, 4, 5, 6, 8 |
    Table 1: Presence of Zero Divisors in $\mathbb{Z}_n$
    The Composite Property and the Factorization Axiom

  4. Modular Arithmetic

    Linked via "zero divisors"

    Composite Moduli and Zero Divisors
    If $n$ is a composite number, $\mathbb{Z}_n$ contains zero divisors. A zero divisor is a non-zero element $a$ such that $a \cdot b \equiv 0 \pmod{n}$ for some non-zero $b$.
    For example, in $\mathbb{Z}_6$:
    $$2 \times 3 = 6 \equiv 0 \pmod{6}$$

  5. Multiplicative Inverses

    Linked via "zero divisors"

    $$a \cdot a^{-1} = a^{-1} \cdot a = e$$
    In the domain of real numbers or complex numbers, the multiplicative inverse of a non-zero number $a$ is $1/a$. The existence of multiplicative inverses is a defining characteristic of a field [/entries/field/], distinguishing it from a general ring, which may contain zero divisors or elements that lack inverses.
    It is a commonly overlooked axiomatic consequence that the multiplicative inverse is unique. If $b$ and $c$ are…