m-Formally Noetherian/Artinian rings

The purpose of this paper is to introduce two new classes of rings that are closely related to the classes of Noetherian rings and Artinian rings. Let R be a commutative unitary ring and m a positive integer. We call R to be m-formally Noetherian (respectively m-formally Artinian) if for every increasing (respectively decreasing) sequence (In)n0 of ideals (respectively proper ideals) of R, the increasing (respectively decreasing) sequence (Sigma i1++im=nIi1...Iim)n0 stabilizes. We show that many properties of Noetherian (respectively Artinian) rings are also true for m-formally Noetherian (respectively m-formally Artinian) rings and we give many examples of m-formally Noetherian/Artinian rings. We investigate the m-formally variant of some well known theorems on Noetherian and Artinian rings. We prove that R is m-formally Artinian for some m if and only if R is m-formally Noetherian for some m with zero Krull dimension. We show that the m-formally variant of Eakin-Nagata theorem is true in some way in the zero-dimensional case.