Can somebody explain me this proof on Kasch book (Modules and rings)? I have a question about one step in the proof of Proposition 13.2.6:
If $R_R$ is injective and ${_R}R$ is noetherian then $R_R$ is a cogenerator and $R$ is artinian on both sides.
In the proof of this Proposition Kasch deduces that:


*

*$Rad(R)$ is nilpotent.

*$\bar{R}=R/Rad(R)$ is regular.

*$\bar{R}$ is semisimple.


I understand these three points, but the next step is confuse to me. Te next affirmation is "Consequently by 11.6.3, $R$ is perfect on both sides".
The Proposition 11.6.3 is the Bass' Theorem P, thah states that the following conditions are equivalent for a ring $R$:
a) $R$ is right perfect.
b) Every flat right $R$-module is projective.
c) $R$ satisfies the descending chain condition for cyclic left ideals.
d) Every left $R$-module $\neq 0$ possesses a socle $\neq 0$ and $R$ contains no infinite set of orthogonal idempotents.
e) $R/Rad(R)$ is semisimple and $Rad(R)$ is left t-nilpotent.
For Kasch, $R$ is left t-nilpotent if for every family $a_1,a_2,a_3,...$ of elements in $R$, exists $k\in \mathbb{N}$ with
$a_ka_{k-1} \cdots a_1=0$.
Can anybody help me with this, please? Thank you.

A nilpotent ideal is $t$-nilpotent on both sides, so the final set of criteria for perfectness applies.
Part of the problem may be reflected in the fact you didn't write the definition of left t-nilpotent correctly. The sequence of elements is taken from within $J(R)$.
