Let be a ring. Let .
The ideal quotient of by is
We call
the annihilator of , also denoted
by .
If we write and .
Using this notation, we see that the set of zero divisors of
is precisely the set .
Example:
Take . Let .
Write
for some primes and nonnegative exponents
.
Then
where and
In other words, where .
The following are easily verified.
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If for then
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If for then