An extension ofz-ideals andz◦-ideals

Abstract

LetRbe a commutative with unity,Y⊆Spec(R), and $h_Y$(S) ={P∈Y:S⊆P}, foreveryS⊆R. An idealIis said to be an $H_Y$-ideal whenever it follows from $h_Y$(a)⊆ $h_Y$(b)anda∈Ithatb∈I. A strong $H_Y$-ideal is defined in the same way by replacingan arbitrary finite setFinstead of the elementa. In this paper these two classes ofideals (which are based on the spectrum of the ring Rand are a generalization of thewell-known concepts semiprime ideal, z-ideal,z^circ-ideal (d-ideal), sz-ideal and s z^circ-ideal (ξ-ideal)) are studied. We show that the most important results about these concepts, "Zariskitopology", "annihilator", and etc. can be extended in such a way that the correspondingconsequences seems to be trivial and useless. This comprehensive look helps to recognizethe resemblances and differences of known concepts better.