Centrally Prime Rings which are Commutative

In this paper the definition of centrally prime rings is introduced , our main purpose is to classify those centrally prime rings which are commutative and so that several conditions are given each of which makes a centrally prime ring commutative. The Fundamentals Let R be a ring .A non-empty subset S of R is said to be a multiplicative closed set in R if S b a , implies S ab , and a multiplicative closed set S is called a multiplicative system if S 0 ,(Larsen &McCarthy ,1971).Let S be a multiplicative system in R such that } 0 { ] , [ R S ,where } , : ] , {[ ] , [ R r S s r s R S and ] , [ r s is the commutator defined by rs sr . Define a relation (~) on S R as follows : If 0 ) ( ) , ( ~ ) , ( ) , ( ), , ( bs at x that such S x iff t b s a then S R t b s a Since } 0 { ] , [ R S ,it can be shown that (~) is an equivalence relation on S R .Now denote the equivalence class of ) , ( s a in S R by s a , that is )} , ( ~ ) , ( : ) , {( t b s a S R t b s a (this equivalence class is also denoted by s a (Larsen & McCarthy,1971) or by a s 1 , and then denote the set of all equivalence classes determined under this equivalence relation by S R , that is let } ) , ( : { S R s a s a S R .Note that S R is also denoted by R S 1 (Larsen & McCarthy,1971 ;Ranicki,2006). On S R we define addition ) ( and multiplication (.) as follows: st bs at t b s a ) ( . S R t b s a st ab t b s a , , ) ( . .