On (semi)regularity and the total of rings and modules. (Q731232)

In a series of recent works (Kasch, Mader, Beidar, Schneider, Zhou) some notions of rings and modules were generalized for the group of homomorphisms \(\hom_R(M,N)=[M,N]\). In particular, some substructures of \([M,N]\) were defined and studied as: Jacobson radical, singular ideal, cosingular ideal and total. The basic notions in these investigations are regularity and semiregularity. This paper contains some results on these substructures. The relation \(\text{Tot}[M,N]=J[M,N]\) is true if and only if \([M,N]\) is semipotent. For \(M=\bigoplus M_i\) and \(N=\bigoplus N_j\), \([M,N]\) is semipotent if and only if \([M_i,N_j]\) is semipotent for all \(i\) and \(j\). Let \(M=\bigoplus M_i\) and \(N=\bigoplus N_j\) be LE-decompositions of modules. Then \([M,N]\) is semipotent if and only if \([M,N]\) is semiregular. By conditions of type \(\text{Tot}[M,N]=J[M,N]\) the semisimple Artinian rings are characterized. Every ring \(R\) has a unique maximal semiregular ideal \(M'(R)\) and \(M'\) has some ``radical'' properties. A series of examples helps to discuss some open questions in this domain.