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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 $$(I_n)_{n\ge 0}$$ ( I n ) n ≥ 0 of ideals (respectively proper ideals) of R, the increasing (respectively decreasing) sequence $$(\sum _{i_1+\cdots +i_m=n}I_{i_1} \ldots I_{i_m})_{n\ge 0}$$ ( ∑ i 1 + ⋯ + i m = n I i 1 … I i m ) n ≥ 0 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 $$m'$$ m ′ -formally Noetherian for some $$m'$$ 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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