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Paper   IPM / M / 9538
School of Mathematics
  Title:   Strong zero-divisors of non-commutative rings
  Author(s):  M. Behboodi (Joint with R. Beyranvand)
  Status:   Published
  Journal: J. Algebra Appl.
  Vol.:  8
  Year:  2009
  Pages:   565-580
  Supported by:  IPM
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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