Question.Is there any standard name for a (commutative or non-commutative) unital ring $R$ with the property that, for every $a in R$, the (descending) chain $R, aR, a^2 R, ldots,$ is eventually constant?

Let me refer to this condition as the DCCPRP, that is, the “DCC on chains of principal right ideals generated by the powers of an element”.

The DCCPRP is satisfied by *left* perfect rings, which, by a famous theorem of Bass (Theorem 28.4 in Anderson & Fuller’s book), can be equivalently characterized as the rings satisfying the DCC on principal right ideals (DCCPR). However, the DCCPRP is much weaker than the DCCPR.

E.g., let $R$ be the commutative, boolean ring with identity obtained by endowing the power set $P(X)$ of a set $X$ with the operations of symmetric difference (as addition) and intersection (as multiplication). The DCCPRP is trivially verified in this case (as it would be in any boolean ring), while the DCCPR holds if and only if $X$ is finite: For, suppose $X$ is infinite. Accordingly, let $x_1, x_2, ldots$ be an infinite sequence of pairwise distinct elements from $X$, and set $X_0 := X$ and $X_k := X setminus {x_1,ldots, x_k}$ for each $k in mathbf N^+$. Then $X_0 P(X), X_1 P(X), ldots$ is a strictly decreasing sequence of principal right ideals of $R$, because $X_k P(X) = P(X_k)$ for every $k in mathbf N$ and, by construction, $X_{k+1} subsetneq X_k$.