158 Journal of Kirkuk University –Scientific Studies , vol.3, No.1,2008 On Centrally Semiprime Rings and Centrally Semiprime Near-Rings with Derivations Adil Kadir Jabbar and Abdulrahman Hamed Majeed College of Science - University of Sulaimani College of Science - University of Baghdad Abstract In this paper, two new algebraic structures are introduced which we call a centrally semiprime ring and a centrally semiprime right near-ring, and we look for those conditions which make centrally semiprime rings as commutative rings, so that several results are proved, also we extend some properties of semiprime rings and semiprime right near-rings to centrally semiprime rings and centrally semiprime right near-rings. Introduction Let R be a ring . A non-empty subset S of R is said to be a multiplicative system in R if S 0 and S b a , implies S ab(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 Ras follows : If S R t b s a ) , ( ), , ( , then ) , ( ~ ) , ( t b s a if and only if there exists S x such that 0 ) ( bs at x . Since } 0 { ] , [ R S , it can be shown that (~) is an equivalence relation on S R. Let us denote the equivalence class of ) , ( s a in S Rby s a , and the set of all equivalence classes determined under this equivalence relation by S R , that is, let )} , ( ~ ) , ( : ) , {( t b s a S R t b s a and } ) , ( : { S R s a s a S R . (the equivalence class s a is also denoted by s a (Larsen & McCarthy, 1971) or by a s 1 (Ranicki, 2006), and S R is also denoted by R S 1 (Larsen & McCarthy, 1971 ; Ranicki, 2006). We define addition ) ( and multiplication (.) on S R as follows: st bs at t b s a ) ( and st ab t b s a ) ( . , for all S R t b s a , . It can be shown that these two operations are well-defined and that ,.) , ( S R forms a ring which is known as the localization of R at S ,(Fahr, 2002).