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Paper IPM / M / 8284  


Abstract:  
In a commutative ring R, an ideal I consisting entirely of
zero divisors is called a torsion ideal, and an ideal is called a
z^{o}ideal if I is torsion and for each a ∈ I 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 zideal is a z^{O}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 zideal) is a z^{O}ideal, if and only if every torsion
zideal is a z^{O}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 z^{O}ideal to
be either minimal or maximal. We show that in a large class of
rings, the sum of two z^{O}ideals is either a z^{O}ideal or R
and we also give equivalent conditions for R to be a PPring
or a Baer ring.
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