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Proof

(related to Proposition: Cancellation Law)

"\(\Leftarrow\)"

Let \(a=b\). Multiplying the equation with \(c\) results in \(ac=bc\).

"\(\Rightarrow\)"

Let \(ac=bc\) and \(c\neq 0\). Following the distributivity axiom in \(R\), we get \((a-b)c=0\). Because \(R\) is an integral domain and \(c\neq 0\), it follows that \((a-b)=0\), which is equivalent with \(a=b\).

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References

Bibliography

1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013