The Three Types Generalized Study On The Rings Of Armendariz

The concept of Armendariz rings was firstly put forward by Rege and Chhawchharia. Let R be a ring, if for anyf(x)=(?)aixi,g(x)=(?)bjxj?R[x]\{0},f(x)g(x)=0leads aibj=0 for all i,j,then the ring R is called an Armendariz ring. In 1974, Armendariz proved that reduced ring was Armendariz ring.In 1998, Anderson and camillo gave more profound results on Armendariz rings. Lately, there are much research results published about Armendariz rings every year, and base on these work, this paper also give three types study of Armendariz rings.Fristly, we introduce the historical background, development pro-cess and research, and briefly summarize some mainly results in the literature, and following results we obtained :1.If R is a weakly 2-primal? rings, then R is a weakly zip ring only and if only R[x] is a weakly zip ring; 2.If R is a weakly 2-primal ? rings, and nil(R) is a ideal of R, then R is a weakly APP-ring only and if only R[x] is a weakly APP-ring;3.If R is a weakly 2-primal ? rings, and nil(R) is a ideal of R, then R is a nilpotent p.p.-ring only and if only R[x] is a nilpotent p.p.-ring.Secondly, We introduce the conception of generalized nil-?-skew Armendariz rings and generalized central ?-Armendariz rings, discuss the properties and characterization of nil-?-skew Armendariz rings and central Armendariz rings, which under the generalized meaning, and following results we obtained :1. Let R, S be rings, a : R? S is a monomorphism, and ??= ??.if S is a generalized nil-?-skew Armen-dariz ring, then R is a generalized nil-?-skew Armendariz ring. 2. Let I be a Nil-ideal of R, and ?(I)(?) I, then R is a generalized nil-?-skew Armendariz ring only and if only R/I is a generalized nil-?-skew Armendariz ring.Finally, We introduce the conception of generalized central ?-Armendariz rings, and by counter-example to explain that generalized central ?-Armendariz ring is not necessarily a-weak Armendariz ring.We also obtained the following results: 1.Let ? be a monomorphism of ring R, and for arbitrary idempotent element e, ?(e) = e. If R is a generalized central a-Armendariz ring, then R is a abeilan ring; 2. R is a generalized central ?-Armendariz rings only and if only ?-R is a generalized central ?-Armendariz ring.