Semipotent rings and modules

This book is an introductory text to the theory of semipotentness of rings and modules which is introduced as a natural generalization of Von Neumann regular rings. In the study of the regular rings, one of the interesting questions is the following: In a regular ring, is every one sided ideal be regular ring?. The answer of this question is negative in general. It was proved that every one sided ideal of a regular (semipotent) ring is semipotent. R. Ware proved that, there are projective regular modules which do not have a regular endomorphism ring. It was proved that, the endomorphism ring of every regular module (in the sense of Zelmanowitz) is semipotent with zero radical Jacobson. The second part of this book is specialize to study of the total, which is first introduced by F. Kasch, and the connection between semipotentness and the total. In addition to that it studies the semipotentness of (M,N) which is introduced as a generalization of the semipotent (endomorphism) ring and the semipotent (endomorphism) ring related to (co)singular ideal. The third part specialize to study of modules which are introduced as a generalization of the regular and locally projective modules.