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

Karim Samei¹

¹ Department of Mathematics Bu Ali Sina University Hamedan, Iran and Institute for Studies in Theoretical Physics and Mathematics (IPM) Tehran, Iran

Fundamenta Mathematicae, Tome 189 (2006) no. 2, pp. 99-109

Résumé

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.

Détail

DOI : 10.4064/fm189-2-1

Keywords: commutative ring ideal consisting entirely zero divisors called torsion ideal ideal called ideal torsion each intersection minimal prime ideals containing contained prove large classes rings say following results every z ideal ideal only every element either zero divisor unit only every maximal ideal general every prime z ideal ideal only every torsion z ideal ideal and only sum torsion ideals either torsion ideal necessary sufficient condition every prime ideal either minimal maximal large class rings sum ideals either ideal equivalent conditions pp ring baer ring