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2006 | 189 | 2 | 99-109
Tytuł artykułu

z⁰-Ideals and some special commutative rings

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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 z⁰-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 z⁰-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 z⁰-ideal, if and only if every torsion z-ideal is a z⁰-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 z⁰-ideal to be either minimal or maximal. We show that in a large class of rings, the sum of two z⁰-ideals is either a z⁰-ideal or R and we also give equivalent conditions for R to be a PP-ring or a Baer ring.
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  • Department of Mathematics, Bu Ali Sina University, Hamedan, Iran
  • Institute for Studies, in Theoretical Physics and Mathematics (IPM), Tehran, Iran
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bwmeta1.element.bwnjournal-article-doi-10_4064-fm189-2-1
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