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

Let M and N be two modules over a ring R. Recent works by Kasch, Schneider, Beidar, Mader, and others have shown that some of the ring and module theory can be carried out in the context of homR(M,N). The study of substructures of homR(M,N) such as the radical, the singular and co-singular ideals and the total has raised new questions for research in this area. This paper is a continuation of study of these substructures, focusing on when the total is equal to the radical, as well as their connections with (semi)regularity of homR(M,N). New results obtained include necessary and sufficient conditions for the total to equal the radical, a description of the maximal regular sub-bimodule of homR(M,N), the existence of the maximal semiregular ideal of a ring, and answers to a number of existing open questions.