The following proposition is only a generalization of a cancelling rule when multiplying integers. In this sense, the ring of integers $(\mathbb Z, + ,\cdot)$ is only one special case of an integral domain, in which this law always holds.

Proposition: Generalization of Cancellative Multiplication of Integers

Let $(R, + ,\cdot)$ be an integral domain. Then the multiplication operation "$\cdot$" is cancellative, i.e. for all $a,b,c\in R$ with $c\neq 0$ the following equivalence holds: $$ac=bc\Longleftrightarrow a=b.$$

Proofs: 1

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  1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013
  2. Kramer Jürg, von Pippich, Anna-Maria: "Von den natürlichen Zahlen zu den Quaternionen", Springer-Spektrum, 2013