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# Show that a commutative ring with the cancellation property (under multiplication) has no zero-divisors. # Show that a commutative ring with the cancellation property (under multiplication) has no zero-divisors.

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Commutative Algebra asked 2020-12-17
Show that a commutative ring with the cancellation property (under multiplication) has no zero-divisors.

## Answers (1) 2020-12-18

Let R be a commutative ring with cancellation property bc=cd
implies $$nb=d$$
for all $$\displaystyle{b},{c},{d}\in{R}$$
If $$cd=0$$ and c in non zero, then
$$cd=c \cdot 0$$
$$d=0$$
by cencellation property.
Since $$cd=c \cdot 0$$
$$d=0$$
Since c and d were arbitary R has non zero divisors

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