Rings which are noetherian modulo a T-nilpotent nilradical

Let [Formula: see text] be a commutative ring with an identity. The ring [Formula: see text] is called N-perfect if it is noetherian modulo its nilradical [Formula: see text] and [Formula: see text] is T-nilpotent. We show that a ring [Formula: see text] is N-perfect if and only if the polynomial ring [Formula: see text] is N-perfect. An [Formula: see text]-module [Formula: see text] is called locally N-perfect if [Formula: see text] is an N-perfect ring for all [Formula: see text]. Several examples of N-perfect rings and locally N-perfect modules are provided. We obtain a characterization of locally N-perfect modules. Also, we characterize the rings [Formula: see text] for which the class of locally artinian (locally noetherian or locally N-perfect) [Formula: see text]-modules is closed under module extensions.