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.

Question
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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