$z^0$-Ideals and some special commutative rings
Volume 189 / 2006
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.
