On graded-nonnil-Noetherian commutative rings

Abstract

Our present work is motivated by several papers which have introduced and studied the notion of nonnil-Noetherian rings. Let A = ⊕ _{α ∈G} A _α be a commutative ring with unity graded by an arbitrary commutative monoid G. We define A to be graded-nonnil-Noetherian ring if every nonnil graded ideal of A, that is, a graded ideal which is not contained in Nil(A), is finitely generated. The purpose of this paper is to give the graded version of several characterizations, properties and basic results proved in [3], [7] and [4]. We also show the non-triviality of our generalization by giving examples of graded-nonnil-Noetherian rings which are not nonnil-Noetherian and of graded-nonnil-Noetherian rings which are not graded-Noetherian.