On Centrally Semiprime Rings with Derivations: Algebra in Mathematics

It is known that a ring R is called a centrally semiprime ring if the localization RS of R is a semiprime ring at each central multiplicative system S of R. We prove that a semiprime ring is a centrally semiprime ring and we give an example for a centrally semiprime ring which is not semiprime and which establishes that centrally semiprime rings are generalizations of semiprime rings. The main aim of this work is to look for those properties of semiprime rings which can be extended to centrally semiprime rings and to find so many conditions under which we can do these extensions and we look for those properties of functions which are preserved under localization and many properties of centrally semiprime rings are proved. Also, we study centralizers and Jordan centralizers of centrally semiprime rings and we find some properties of each one and determining the relationships of Jordan centralizers with strong commutatively preserving mappings of Lie ideals of centrally semiprime rings. The derivations of centrally semiprime rings are studied and we have obtained some properties of them and finally, generalized derivations and orthogonal generalized derivations of centrally.