Fully Prime Rings And Gr-π-Coherent Rings

In recently years, some good results about the study of rings and moudles under the ring extension has been obstained. In there we consider ring extensions are crossed products and group graded rings.The concept of fully prime ring was introduced by W. D. Blair and H. Tsutsui in 1994. they chatacterized the structure of fully prime rings, and discussed the behavior of the fully prime condition when passing to related rings. In chapter 1. the related rings we study are crossed products and group graded rings. At the beginning of section 1.2. we explore the fully prime condition of crossed product R G with G finite, and denote briefly fully prime ring as FPR. Some neccessary conditions and sufficient conditions for R G. a crossed product, to be an FPR are given. For example,Theorem 1.2.1 Let R be an FPR, let G be a finite group and let RG be a crossed product, then the crossed product RGis an FPR if and only if the map : L(RG) L(R).P PR. is a one to one onto correspondence between the set of ideals of R G and the set of ideal of R.Theorem 1.2.2 Let R be a ring, let G be a finite group and let R G be a crossed product, then the following are equivalent:(t) the crossed product R G is an FPR;(ii) (a) R is a G-FPR:(b) the map fa : L(R G) G-?R),P