My apologies if this question is below the level of MO. I posted the [same question](https://math.stackexchange.com/questions/4507690/dcc-on-the-powers-of-ideals) 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 $I=xR$.



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> 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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DCC on the powers of ideals
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**](http://dx.doi.org/10.1016/j.jalgebra.2007.07.016), J. Algebra 319, No. 7, 3006-3027 (2008). [ZBL1168.13002](https://zbmath.org/?q=an:1168.13002).


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


