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 (∈, ∈)- 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.
['Bin Yu', 'Xue-hai Yuan']
The intuitionistic fuzzy subrings and fuzzy ideals