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We introduce the set $S(R)$ of ``strong zero-divisors" in a ring $R$ and
prove that: if $S(R)$ is finite, then $R$ is either finite or a prime ring. When certain sets of ideals
have $ACC$ or $DCC$, we show that either $S(R)=R$ or $S(R)$ is
a union of prime ideals each of which is a left or a right
annihilator of a cyclic ideal. This is a finite union when $R$
is a Noetherian ring. For a ring $R$ with $|S(R)|=p$, a prime
number, we characterize $R$ for $S(R)$ to be an ideal.
Moreover $R$ is completely characterized when $R$ is a ring with identity and $S(R)$ is an ideal with $p^2$ elements.
We then consider rings $R$ for which $S(R)=Z(R)$, the set of zero-divisors,
and determine strong zero-divisors of matrix rings over
commutative rings with identity.
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