 Advances in Pure Mathematics, 2012, 2, 217-219 http://dx.doi.org/10.4236/apm.2012.24031 Published Online July 2012 (http://www.SciRP.org/journal/apm) On Commutativity of Semiprime Right Goldie Ck-Rings Najat M. Muthana, S. Khalid Nauman Department of Mathematics, King Abdulaziz University, Jeddah, KSA Email: {najat_muthana, synakhaled}@hotmail.com Received December 2, 2011; revised February 5, 2012; accepted February 14, 2012 ABSTRACT This short exposition is about some commutativity conditions on a semiprime right Goldie Ck-ring. In particular, it is observed here that a semiprime right Goldie Ck-ring with symmetric quotient is commutative. The statement holds if the symmetric ring is replaced by reduced, 2-primal, left duo, right duo, abelian, NI, NCI, IFP, or Armendariz ring. Keywords: Semiprime Right Goldie Ck-Rings; Reduced; Symmetric; Von Neumann Regu lar Rings In this short note we expose some commutativity condi- tions on a semiprime right Goldie Ck-ring. All rings here are associative with an identity. A ring A is said to be a Ck-ring, as introduced by Chuang and Lin in 1989 in , if for very pair of elements x, y  A, there exist integers m = m(x, y) and n = n(x, y) such that ,0mnkxy, where [x, y]k is the kth-commutator defined by Klein, Nada, and Bell in  in 1980, as 1,,kk,xyxyy where 1,,xyxy. A ring is called a symmetric ring (in the sense of Lam- bek ), if when ever rab = 0, then rba = 0, , semiprime (respt. reduced) if A has no non-zero nilpotent ideal (respt. element) and von Neumann regular if for each a ,,rabA A, there exists r  A such that ara = a. A ring is right Goldie in case it has finite right uniform dimen- sion and satisfies acc on right annihilators. In [1; Theorem 1] Chaung-Lin proved that: Lemma 1: Every reduced Ck-ring is commutative. We use it to prove the following. Theorem: A semiprime right Goldie Ck-ring with sym- metric right quotient is commutative. Proof: Lambek in [3; Section 1G] proved that every reduced ring is symmetric. We prove that the converse holds for von Neumann regular rings. In deed, one may deduce easily that A is symmetric if and only if 12 0kaa a, then   12 0pp pkaaa , where ai, ap(i)  A and p is a one-to-one correspondence on the set {1, 2, ···, k}. Let a  N(A) be a non-zero ele- ment of some index n. Since A is von Neumann regular, for some x  A, 1naaxa axa . But A is symmetric and an = 0, which implies that a = anxn−1 = 0. Hence A is reduced. The famous Goldie’s Theorem states that a ring A is semiprime right Goldie if and only if A has a right quo- tient ring B which is semisimple Artinian [4; Theorem 2.3.6]. But a semisimple Artinian ring is von Neumann regular [5; Theorem 1.7]. Since B is symmetric and now von Neumann regular, therefore B is reduced. This means that A is reduced. Since A is a Ck-ring, by the Lemma 1, we get that A is commutative. ∎ The statement of the Theorem remains unchanged if we replace the condition of the ring being symmetric by 2-primal, abelian, left or right duo, NI, NCI, IFP, quasi- IFP, near-IFP, Armendariz, weak-Armendariz, and some other relations that are listed in Lemma 2. Let us denote by N(A) the set of all nilpotent elements of A. For a reduced ri ng N(A) = 0, and a ring is NI if N(A) is an ideal , NCI if N(A) contains a nonzero ideal , and 2-primal if N(A) is the intersection of prime ideals . A ring A is said to have “Insertion of factor property (in short, IFP)  in case for any pair of elements a, b of A, if ab = 0, then arb = 0 for all . Such rings are also termed as semicommutative in literature, we simply call them IFP rings. Near-IFP (respt. quasi-IFP) rings are introduced recently in  (respt. in ), and are characterized asAaA contains a non-zero nilpotent ideal of A for any rA0aA in [10; Proposition 1.2] (respt. AaA is a nilpotent ideal of A for any in [11; Lemma 1.3]). 0aABy definitions, every reduced ring is an IFP ring, an IFP ring is a quasi-IFP ring, and a quasi-IFP ring is a Copyright © 2012 SciRes. APM N. M. MUTHANA, S. K. NAUMAN 218 near-IFP ring. The converse need not be true in general (see the Example below) but for a semiprime ring it holds. A ring A is called Armendariz in  if whenever polynomials in Ax, 01 mmfxaax ax n and n01gxbb bxx 0fxgx satisfy , then for each i, j, and weak Armendariz in  if whenever , then for each i, j, . 0ijab 01aax01 0bbx0ijab For several interactions and various characterizations with examples and counter examples of these rings which we have discussed above the interested reader may refer to the articles [4,11,12,14,15]. Example: It is clear from the examples and counter examples in above citations that the rings listed above are different from each other, but we found no example for near-IFP and quasi-IFP rings to be different in literature. By definitions, quasi-IFP is near-IFP, we prove that the opposite may not be true. Let R be a ring and a nilpotent ideal of R such that every element of is a unit. For example, a local ring is of this type. By Proposition 1.10  is near-IFP. Let. It is clear that 0IRInnRMatR0ij nrNRnijnnRrR R is nilpotent if , otherwise not nilpotent in general. The ideal n of n is proper and nilpotent. Hence we conclude that is not quasi-IFP. ∎ ijrInnat IRMRRnNow we give a list of rings which coincide on the condition of von Neumann regularity. Here P stands for some property, for instance, property for being reduced, etc. RLemma 2: Let A be a von Neumann regular ring. Then the following are equivalent. (P1) A is reduced; (P2) A is left (or right) duo; (P3) A is abelian; (P4) A is 2-primal; (P5) A is symmetric; (P6) A is NI; (P7) A is NCI; (P8) A is IFP; (P9) A is quasi-IFP; (P10) A is near-IFP; (P11) A is a subdirect product of division ring; (P12) A is Armendariz; (P13) A is weak Armendariz; (P14) If such that with , then ,, ,aaaA0.aa a20aa a 21n(P15) If such that , then ,, ,aaaA0aa a 0.aa aProof: The equivalence (P1) (P5) is proved in the Theorem above. Equivalences of (P1)-(P4) and (P6) and (P7) hold in [11; Proposition 1.4]. It is clear by defini-tions that every reduced ring is IFP, every IFP ring is quasi-IFP and every quasi-IFP ring is near IFP. Thus (P1) (P8)  (P9)  (P10). Because a von Neumann regular ring is semiprime, by [10; Proposition 1.4], for a semiprime ring a near-IFP ring is reduced, giving the equivalence (P1)  (P10). Equivalences of (P1)-(P3), (P6), (P8) and (P10) also hold in [10; Proposition 1.6]. The equivalence of a von Neu-mann regular ring to be reduced, Armendariz, and weak Armendariz is proved in [14; Lemma 2.4]. Finally, the equivalences (P1)  (P2)  (P3)  (P8) (P11)  (P12)  (P13)  (P14)  (P15) are established in [14; Lemma 2.4]. ∎ Lemma 3 Let A be a semiprime ring of bounded index of nilpotency. Then the following conditions are equiva-lent: (P1) A is reduced; (P4) A is 2-primal; (P6) A is NI; (P7) A is NCI; (P8) A is IFP; (P9) A is quasi-IFP. Proof: (P1)  (P4)  (P6)  (P7) hold by [7; Proposition 1.3], (P1)  (P6)  (P8)  (P10) hold by [10; Proposition 1.5] while (P1)  (P4)  (P6)  (P8)  (P9) hold in [11; Proposition 1.6]. ∎ The consequences of the Theorem and above lemmas are the following. Corollary 1: A Ck-von Neumann regular ring is com-mutative if any one of the properties (P1)-(P15) of Lemma 2 is satisfied. Corolla ry 2: A Ck-semiprime ring of bounded index of nilpotency is commutative if any one of the properties (P1), (P4), (P6)-(P9) of Lemma 3 is satisfied. Corollary 3: Let A be a semiprime right Goldie ring and B its classical ring of quotients. Then the ring B sat-isfies all equivalent conditions from (P1) to (P15) of Lemma 2. Moreover, for the ring A, the conditions (P1), (P4), (P8), (P9), (P10), (P12) and (P13) of Lemma 2 are mutually equivalent and are also equivalent to above each of fifteen conditions for the ring B. Proof: Equivalence of (P1) to (P15) is followed from [14; Theorem 2.6] and Lemma 1.2. If A is near-IFP and semiprime, and if a is nilpotent, then every ideal of AaA is zero. Hence, in particular, a is zero, and A is reduced. So, (P1), (P8)-(P10) are equiva-lent for the ring A. 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