“School of Mathematics”
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| Paper IPM / M / 8284 |
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| Abstract: | |
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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 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 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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