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Recall that a subset $I$ of a ring $R$ is left (resp., right) $T$-nilpotent in case for every sequence $$a_1,a_2,\cdots $$ in $I$ there is an $n$ such that $a_1\cdots a_n=0$ (resp., $a_n\cdots a_1=0$). Every nilpotent ideal is left and right $T$-nilpotent.

Is there any characterization for rings $R$ in which the Jacobson radical is $T$-nilpotent?

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    $\begingroup$ I suppose you are well aware of this, but just in case other readers might be, like me, a little forgetful, a ring is left-perfect iff $J(R)$ is left T-nilpotent and $R/J(R)$ is semisimple, i.e., left-Artinian (where $J(R)$ is the Jacobson radical). $\endgroup$
    – Gro-Tsen
    Jan 25, 2016 at 15:30
  • $\begingroup$ Yes, I've seen that before but I don't know exactly where:) Could you please give a reference for this fact.? And since for most of the rings (i.e. left/right Artinian) one has that R/J(R) is semisimple it seems that the answer would be that J(R) is $T$-nilpotent if and only $R$ is left perfect. $\endgroup$
    – Sebastian Burciu
    Jan 25, 2016 at 16:08
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    $\begingroup$ See Lam, A First Course in Noncommutative Rings (2d edition Springer 2001, GTM 131), §23 ("Perfect and Semiperfect Rings"), esp. around definition 23.18 (of left-perfect rings) and theorem 23.20 (Bass criterion). $\endgroup$
    – Gro-Tsen
    Jan 25, 2016 at 16:14

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