Abstract
In this paper, centrally prime and centrally semiprime rings are defined and the relations between these two rings and prime (resp. semiprime) rings are studied.Among the results of the paper some conditions are given under which prime (resp. semiprime) rings become centrally prime (resp.centrally semiprime) as in:1-A nonzero prime (resp. semiprime) ring which has no proper zero divisors is centrally prime (resp.centrally semiprime).Also we gave some other conditions which make prime (resp. semiprime) rings and centrally prime (resp.centrally semiprime) rings equivalent, as in :2-A ring which satisfies the- ) (BZP for multiplicative systems is prime (resp. semiprime) if and only if it is centrally prime (resp.centrally semiprime).3-A ring with identity in which every nonzero element of its center is a unit is prime (resp. semiprime) if and only if it is centrally prime (resp.centrally semiprime).
Highlights
E (resp.centrally semiprime). we gave some other conditions which make prime (resp. semiprime) rings and centrally prime (resp.centrally semiprime) rings equivalent, as in :2-A ring which satisfies the- (BZP) for multiplicative systems is prime (resp. semiprime) if and only if it is centrally prime (resp.centrally semiprime).3-A ring with identity in which every nonzero element of its center is a unit is prime (resp. semiprime) if and only if it is centrally prime (resp.centrally semiprime)
E. we gave some other conditions which make prime rings and centrally prime rings equivalent, as in :2-A ring which satisfies the- (BZP) for multiplicative systems is prime if and only if it is centrally prime.3-A ring with identity in which every nonzero element of its center is a unit is prime if and only if it is centrally prime
To prove the “if”part, let as RS bt = {0}, where a, b R and s, t S, for any r R we have rs RS, and asrsbt as RS bt, which gives asrsbt = 0 orsst = 0, there exists t S ( t depends on r) such that t(arb) = 0, arb ann(t) and S bieng a bi-zero multiplicative system so ann(t) = 0 and we get arb = 0, this last result is true for all r R, which implies that aRb = {0}, and this completes the proof
Summary
E (resp.centrally semiprime). we gave some other conditions which make prime (resp. semiprime) rings and centrally prime (resp.centrally semiprime) rings equivalent, as in :2-A ring which satisfies the- (BZP) for multiplicative systems is prime (resp. semiprime) if and only if it is centrally prime (resp.centrally semiprime).3-A ring with identity in which every nonzero element of its center is a unit is prime (resp. semiprime) if and only if it is centrally prime (resp.centrally semiprime). To prove the “if”part, let as RS bt = {0} , where a, b R and s, t S , for any r R we have rs RS , and asrsbt as RS bt , which gives asrsbt = 0 or (arb)sst = 0 , there exists t S ( t depends on r) such that t(arb) = 0 , arb ann(t) and S bieng a bi-zero multiplicative system so ann(t) = 0 and we get arb = 0 , this last result is true for all r R , which implies that aRb = {0}, and this completes the proof .
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