on higher order z-ideals and z◦-ideals in commutative rings

A ring r is called radically z-covered (resp. radically z◦-covered) if every √ zideal (resp. √ z◦-ideal) in r is a higher order z-ideal (resp. z◦-ideal). in this article we show with a counter-example that a ring may not be radically z-covered (resp. radically z◦-covered). also a ring r is called z◦-terminating if there is a positive integer n such that for every m ≥ n, each z◦m-ideal is a z ◦n-ideal. we show with a counter-example that a ring may not be z◦-terminating. it is well known that whenever a ring homomorphism φ : r → 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 φ −1[q] = p), and if r is a z ◦-terminating ring or radically z◦-covered ring then so is s. we prove that a surjective ring homomorphism φ : r → s is strong if and only if ker(φ) ⊆ rad(r).