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“School of Mathematics”

Paper   IPM / M / 8284
   School of Mathematics
  Title: Z0-ideals and some special commutative rings
  Author(s): K. Samei
  Status: Published
  Journal: Fund. Math.
  Vol.: 189
  Year: 2006
  Pages: 99-109
  Supported by: IPM
In a commutative ring R, an ideal I consisting entirely of zero divisors is called a torsion ideal, and an ideal is called a zo-ideal if I is torsion and for each aI the intersection of all minimal prime ideals containing a is contained in I. We prove that in large classes of rings, say R, the following results hold: every z-ideal is a zO-ideal if and only if every element of R is either a zero divisor or a unit, if and only if every maximal ideal in R (in general, every prime z-ideal) is a zO-ideal, if and only if every torsion z-ideal is a zO-ideal and if and only if the sum of any two torsion ideals is either a torsion ideal or R. We give a necessary and sufficient condition for every prime zO-ideal to be either minimal or maximal. We show that in a large class of rings, the sum of two zO-ideals is either a zO-ideal or R and we also give equivalent conditions for R to be a PP-ring or a Baer ring.

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