Problem 1 and Problem 3 in Auslander - ReIten and Smalo 's book deal with 

the isomorphism problem and the extension problem for uniserial modules..

In this talk I will present the partial solution to these problems obtained in

[ 1 ]  and  [ 2 ] .  Concerning the isomorphism problem, under the hypothesis 

that  L  and  M  are two uniserial modules over any ring, with well behaved 

lattices of their submodules, we prove a "two fixed point theorem".

Concerning the extension problem for uniserial modules, we prove a 

"two terms in the middle theorem ". 

[ 1 ]  G. D'este - F. Kaynarca - D. Keskin Tutuncu, Isomorphism problem

         for uniserial modules over arbitrary rings, Comm. Algebra Vol. 48,

         No. 2 (2020)  4027 - 4036 .

[ 2 ] G. D'Este - F. Kaynarca - D. Keskin Tutuncu, Extensions of uniserial modules

        Rend. Sem. Mat. Univ. Padova, Forthcoming articles.