The intuitionistic fuzzy subrings and fuzzy ideals

In this paper,we present the concept of (α, β)-intuitionistic fuzzy subring(ideal). And we show that, in 16 kinds of (α, β)- intuitionistic fuzzy subrings(ideals), the significant ones are the (∈, ∈q)-intuitionistic fuzzy subring(ideal), the (∈, ∈ ⋁q)-intuitionistic fuzzy subring(ideal) and the (∈ ∧q; ∈)- intuitionistic fuzzy subring(ideal). We also show that A is a (∈, ∈)- intuitionistic fuzzy subring(ideal) of R if and only if, for any a ∈ (0, 1], the cut set A_a of A is a 3-valued fuzzy subring(ideal) of R, and A is a (∈,∈ ⋁q)-intuitionistic fuzzy subring(ideal)( or (∈ ∧q, ∈)-intuitionistic fuzzy subring(ideal)) of R if and only if, for any a ∈ (0, 0.5] (or for any a ∈ (0.5,1]), the cut set A_a of A is a 3-valued fuzzy subring(ideal) of R. At last, we generalize the (∈,∈)- intuitionistic fuzzy subring(ideal), the (∈,∈ ⋁q)-intuitionistic fuzzy subring(ideal) and the (∈ ∧q, ∈)- intuitionistic fuzzy subring(ideal) to intuitionistic fuzzy subring(ideal) with thresholds, i.e.,(s, t]- intuitionistic fuzzy subring(ideal). We show that A is a (s, t]- intuitionistic fuzzy subring(ideal) of R if and only if, for any a ∈ (s, t], the cut set A_a of A is a 3-valued fuzzy subring(ideal) of R. We also characterize the (s, t]- intuitionistic fuzzy subring(ideal) by the neighborhood relations between a fuzzy point x_a and an intuitionistic fuzzy set A.