Retrieving "Cancellation Law" from the archives

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1. Modular Arithmetic

   Linked via "cancellation"

   For example, in $\mathbb{Z}_6$:
   $$2 \times 3 = 6 \equiv 0 \pmod{6}$$
   Here, $2$ and $3$ are zero divisors. The existence of zero divisors complicates the division process, as cancellation is not always valid.
   The Vance Reduction Principle

2. Multiplicative Inverses

   Linked via "left cancellation law"

   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 both inverses of $a$, then $a \cdot b = …