Rostam Mohamadian - Department of Mathematics, Faculty of Mathematical Sciences and Computer, Shahid Chamran University of Ahvaz, Ahvaz, Iran.

A ring R is called radically z-covered (resp. radically z^\circ-covered) if every \sqrt z-ideal (resp. \sqrt {z^\circ}-ideal) in R is a higher order z-ideal (resp. z^\circ-ideal). In this article we show with a counter-example that a ring may not be radically z-covered (resp. radically z^\circ-covered). Also a ring R is called z^\circ-terminating if there is a positive integer n such that for every m\geq n, each z^{\circ m}-ideal is a z^{\circ n}-ideal. We show with a counter-example that a ring may not be z^\circ-terminating. It is well known that whenever a ring homomorphism \phi:R\to S is strong (meaning that it is surjective and for every minimal prime ideal P of R, there is a minimal prime ideal Q of S such that \phi^{-۱}[Q] = P), and if R is a z^\circ-terminating ring or radically z^\circ-covered ring then so is S. We prove that a surjective ring homomorphism \phi:R\to S is strong if and only if {\rm ker}(\phi)\subseteq{\rm rad}(R).

Radically z-covered, Radically z^\circ-covered, z^n-ideal, z^{\circ n}-ideal, z^\circ-terminating