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|Paper IPM / M / 9538||
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 p2 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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