FURTHER PROPERTIES OF THE GENERALIZATION OF PRIMAL SUPERIDEALS

Abstract

In [7], the author studied several generalizations of primal superideals of a commutative super-ring R.  This paper is devoted to study further properties of ϕ-primal superideals of R, where ϕ : \(\mathcal{I}\)(R) \(\longrightarrow\) \(\mathcal{I}\)(R) \(\cup\) {∅} is any function and \(\mathcal{I}\)(R) is the set of all proper superideals of R.  In particular, if J ∈ \(\mathcal{I}\)(R) then there is a one-to-one correspondence between ϕ-P^I-primal superideals I of R with J ⊆ ϕ(I) and ϕ_J-P^I/J-primal superideals of R/J.  Moreover, for a multiplicatively closed subset S of h(R), if I − ϕ(I) = ρ⁻¹(I_S) − ρ⁻¹(ϕ_S (I_S)) for any I ∈\(\mathcal{I}\)(R), then there is a one-to-one correspondence between ϕ-P ^I-primal superideals I of R and ϕ_S-\(P_{S}^{I}\)(R)-primal superideals I_S of R_S with P^I \(\cap\) S = ∅.