On zS-ideals of commutative rings

Let R be a commutative ring with nonzero unity and S a multiplicatively closed subset (m.c.s) of R. An S-disjoint ideal of R is called z(S)-ideal if there exists s = s(I) is an element of S and whenever M-S(a) subset of M-S(b) for some a is an element of I and b is an element of R, then sb is an element of I, where M-S(a) = {Max(S)(R) : sa is an element of M for some s is an element of S}, for each a is an element of R. In the present paper, we introduce and study the concept of z(S)-ideal which is an extension of the notion of the z-ideals. We generalize some well-known results on z-ideal to z(S)-ideal. In addition, many examples, characterizations, and properties of z(S)-ideal are given. We also investigate the relation between z-ideals and z(S)-ideals of a commutative ring.