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My apologies if this question is below the level of MO. I posted the same question in MS about a week ago without an answer so far.

Let $R$ be a unital commutative ring. $R$ is called strongly $\pi$-regular if it satisfies the DCC on the powers of principal ideals, that is, the chain $\hspace{3mm}I\supseteq I^2 \supseteq I^3 \supseteq\dots\supseteq I^n\supseteq\dots\hspace{3mm}$ stops for each $x\in R$, where $I=xR$.

Do we have a term coined for those rings that satisfy DCC on the powers of all maximal ideals (respectively, prime ideals, semiprime ideals, semiprimitive ideals, all ideals) ?

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I am not aware of a term for rings with this property, and I have rarely seen it in the literature. There is a small result about such rings in Proposition 3.22 of the following paper:

Lam, T. Y.; Reyes, Manuel L., A prime ideal principle in commutative algebra, J. Algebra 319, No. 7, 3006-3027 (2008). ZBL1168.13002.

Part of that result says that a commutative noetherian ring satisfying this DCC property on powers of maximal is already artinian.

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  • $\begingroup$ +1 Thank you truly for your reply and for the reference. If I may, what more could we say if $R$ is not only a ring but an operator algebra (resp. Banach algebra) ? It's been known that Noetherian and Artinian Banach algebras are finite dimensional link.springer.com/article/10.1007/BF01344169 . $\endgroup$ Commented Aug 16, 2022 at 12:20

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