Beitr Algebra Geom https://doi.org/10.1007/s13366-018-0404-8 ORIGINAL PAPER Mohamed Khalifa Received: 9 February 2018 / Accepted: 28 May 2018 © The Managing Editors 2018 Abstract 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 (I ) of ideals (respectively proper ideals) of R, the increasing n n≥0 (respectively decreasing) sequence ( I ... I ) stabilizes. We show i i n≥0 i +···+i =n 1 m 1 m 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 Noethe- rian 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.
Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry – Springer Journals
Published: Jun 2, 2018
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