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Proof: By Induction

(related to Proposition: Generalization of Cancellative Multiplication of Integers)

"$\Rightarrow$"

• Assume, $ac=bc.$
• Because of the distributivity law in rings, it follows $(a-b)c=0.$
• By hypothesis, $(R, + ,\cdot)$ is an integral domain and $c\neq 0.$
• Therefore $a-b=0.$
• This is equivalent to $a=b.$

"$\Leftarrow$"

• Assume, $a=b.$
• Multiplication by $c$ gives us $ac=bc.$
  ∎

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References

Bibliography

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