On Centrally Prime and Centrally Semiprime Rings

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).


Introduction:
Let R be a ring .A non-empty subset S of R is said to be a multiplicative closed set in implies that S ab , (Larsen and McCarthy,1971) and a multiplicative closed set S is called a multiplicative system if S  0 , (Larsen and McCarthy ,1971 It is known that this ring is not commutative.Take Similarly it can be shown that .That means it is also possible for non commutative rings to have multiplicative systems with above two properties. Remark: Let R be a ring and S a multiplicative system in R such that (Jabbar,2004 }  8  11  ,  4  11  ,  2  11  ,  1  11  ,...,  8  2  ,  4  2  ,  2  2  ,  1  2  ,  8  1  ,  4  1  ,  2  1  ,  1  1  ,  8  0  ,  4  So that .By the same technique as we used above we can get that and also we will show that this existence becomes unique under certain conditions as we see latter(see Theorem 2 and Theorem 3).Now we mention the following two results the proof of which could be found in (Jabbar,2004). Let R be a ring and S is a multiplicative system in R such that

The Main Results:
First we prove a lemma which will play the basic role ,as we see latter, in the proof of the main results of the paper.

Lemma 1: Let
R be a ring and S a bi-zero multiplicative system in

Proof :
Now let   Proof: The "only if " part has been proved.So it remains to prove the converse part. Let   Let R be a ring.We say that R is centrally prime (resp. centrally semiprime) if S R is prime (resp. semiprime) for all multiplicative systems S in R which have zero commutators.

Example 3:
As we have mentioned in Example 1, that every multiplicative system S in Z is a bi-zero multiplicative system, that is Z satisfies the is a prime ring , and S being arbitrary multiplicative system with zero commutator , so we get that Z is a centrally prime ring .Since every prime ring is a semiprime ring so every centrally prime ring is centrally semiprime and thus Z is also a centrally semiprime ring. Next we apply the result of Lemma 1, to prove some theorems which determine the relations between prime (resp. semiprime) and centrally prime(resp.centrally semiprime) rings, in each of the following two theorems (Theorem 4 and Theorem 5) a condition is given which makes prime (resp. semiprime) rings and centrally prime (resp. centrally semiprime) rings equivalent. Theorem 4 : Let R be a ring. If R satisfies the-) (BZP for multiplicative systems, then R is prime (resp. semiprime) if and only if R is centrally prime (resp. centrally semiprime).

Proof :
Let R be a prime ring and S be any multiplicative system in R which has zero commutator, that is R is prime (resp. semiprime) if and only if it is centrally prime (resp.centrally semiprime) which completes the proof . Now, in each of the following two theorems we give a condition which makes prime (resp. semiprime) rings centrally prime (resp. centrally semiprime),we see below that nonzero prime (resp. semiprime) rings which have no zero divisors are centrally prime (resp. centrally semiprime), which means , in some sense , that centrally prime (resp. centrally semiprime) rings are generalizations of those non-zero rings which have no proper zero divisors.  Next we give another condition under which prime (resp. semiprime) rings are centrally prime (resp. centrally semiprime) and that condition provides R to be a finite ring and this can regarded as a corollary to Theorem 6. Corollary 7 : A finite prime (resp.semiprime) ring R is centrally prime (resp. centrally semiprime).

Proof:
We  . Hence S R is a prime ring which proves that R is centrally prime .
In fact, Corollary 7, tells us that centrally prime (resp.centrally semiprime) rings are, in some sense, generalizations of finite prime (resp. semiprime) rings.