$z^0$-Ideals and some special commutative rings

Volume 189 / 2006

Karim Samei Fundamenta Mathematicae 189 (2006), 99-109 MSC: 13A15, 54C40. DOI: 10.4064/fm189-2-1

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 $z^0$-ideal if $I$ is torsion and for each $a \in I$ 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 $z^0$-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 $z^0$-ideal, if and only if every torsion $z$-ideal is a $z^0$-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 $z^0$-ideal to be either minimal or maximal. We show that in a large class of rings, the sum of two $z^0$-ideals is either a $z^0$-ideal or $R$ and we also give equivalent conditions for $R$ to be a $PP$-ring or a Baer ring.

Authors

  • Karim SameiDepartment of Mathematics
    Bu Ali Sina University
    Hamedan, Iran
    and
    Institute for Studies
    in Theoretical Physics and Mathematics (IPM)
    Tehran, Iran
    e-mail

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