All rings considered in this paper are assumed to be associative with identity 1 6= 0, and all modules are unitary right modules, unless otherwise specified. A ring is called decomposable if it decomposes into a direct product of

two rings, otherwise the ring is indecomposable. The main notions of this paper are that of a prime quiver of a semiperfect

ring and the notion of a prime block. We say that a semiperfect ring A is a prime block if its prime quiver PQ(A) is connected and it is known that

a semiperfect ring A with T -nilpotent prime radical is a prime block if and only if A is indecomposable (see Theorem 4.6). An important class of semiperfect rings is formed by the serial rings.

Artinian serial rings, also called “generalized uniserial rings,” were introduced by Nakayama [21, 22], who showed that all modules over them are serial. The structural description of right Noetherian serial rings was given in [9, 10], where, in particular, the following theorem was proved: