Graded r-Ideals

Let G be a group with identity e and R be a commutative G- graded ring with nonzero unity 1. In this article, we introduce the concept of graded r-ideals. A proper graded ideal P of a graded ring R is said to be a graded r-ideal if whenever a, b ∈ h(R) such that ab ∈ P and Ann(a) ={0}, then b ∈ P. We study and investigate the behavior of graded r-deals to introduce several results. We introduced several characterizations for graded r-ideals; we proved that P is a graded r-ideal of R if and only if aP = aR⋂P for all a ∈ h(R) with Ann(a) = {0}. Also, P is a graded r-ideal of R if and only if P = (P : a) for all a ∈ h(R) with Ann(a) = {0}. Moreover, P is a graded r-ideal of R if and only if whenever A,B are graded ideals of R such that AB ⊆ P and A⋂r(h(R)) ≠ϕ, then B ⊆ P. In this article, we introduce the concept of a huz-rings. A graded ring R is said to be a huz-ring if every homogeneous element of R is either a zero divisor or a unit. In fact, we proved that R is a huz-ring if and only if every graded ideal of R is a graded r-ideal. Moreover, assuming that R is a graded domain, we proved that {0} is the only graded r-ideal of R.