Advances in Pure Mathematics, 2012, 2, 217219 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 CkRings 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 Ckring. In particular, it is observed here that a semiprime right Goldie Ckring with symmetric quotient is commutative. The statement holds if the symmetric ring is replaced by reduced, 2primal, left duo, right duo, abelian, NI, NCI, IFP, or Armendariz ring. Keywords: Semiprime Right Goldie CkRings; Reduced; Symmetric; Von Neumann Regu lar Rings In this short note we expose some commutativity condi tions on a semiprime right Goldie Ckring. All rings here are associative with an identity. A ring A is said to be a Ckring, as introduced by Chuang and Lin in 1989 in [1], if for very pair of elements x, y A, there exist integers m = m(x, y) and n = n(x, y) such that ,0 mn k xy , where [x, y]k is the kthcommutator defined by Klein, Nada, and Bell in [2] in 1980, as 1 ,, kk , yxy y where 1 ,, yxy. A ring is called a symmetric ring (in the sense of Lam bek [3]), if when ever rab = 0, then rba = 0, , semiprime (respt. reduced) if A has no nonzero 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] ChaungLin proved that: Lemma 1: Every reduced Ckring is commutative. We use it to prove the following. Theorem: A semiprime right Goldie Ckring 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 0 k aa a , then 12 0 pp pk aaa , where ai, ap(i) A and p is a onetoone correspondence on the set {1, 2, ···, k}. Let a N(A) be a nonzero ele ment of some index n. Since A is von Neumann regular, for some x A, 1n aaxa 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 C kring, 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 2primal, abelian, left or right duo, NI, NCI, IFP, quasi IFP, nearIFP, Armendariz, weakArmendariz, 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 [6], NCI if N(A) contains a nonzero ideal [7], and 2primal if N(A) is the intersection of prime ideals [8]. A ring A is said to have “Insertion of factor property (in short, IFP) [9] 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. NearIFP (respt. quasiIFP) rings are introduced recently in [10] (respt. in [11]), and are characterized asAaA contains a nonzero 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 quasiIFP ring, and a quasiIFP ring is a C opyright © 2012 SciRes. APM
N. M. MUTHANA, S. K. NAUMAN 218 nearIFP 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 [12] if whenever polynomials in x, 01 m m xaax ax n and n 01 xbb bxx 0fxgx satisfy , then for each i, j, and weak Armendariz in [13] if whenever , then for each i, j, . 0 ij ab 01 aax 01 0bbx 0 ij ab 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 nearIFP and quasiIFP rings to be different in literature. By definitions, quasiIFP is nearIFP, 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 [10] is nearIFP. Let. It is clear that 0I RI nn RMatR 0ij n rNR nijnn RrR R is nilpotent if , otherwise not nilpotent in general. The ideal n of n is proper and nilpotent. Hence we conclude that is not quasiIFP. ∎ ij rI nn at IRMRR n Now 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. R Lemma 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 2primal; (P5) A is symmetric; (P6) A is NI; (P7) A is NCI; (P8) A is IFP; (P9) A is quasiIFP; (P10) A is nearIFP; (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 2 0aa a 2 1n (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 quasiIFP and every quasiIFP 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 nearIFP 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 2primal; (P6) A is NI; (P7) A is NCI; (P8) A is IFP; (P9) A is quasiIFP. 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 Ckvon Neumann regular ring is com mutative if any one of the properties (P1)(P15) of Lemma 2 is satisfied. Corolla ry 2: A Cksemiprime 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 nearIFP 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. Equivalence of (P1) and (P4) for the ring A is obvious and for the same ring (P12) and (P13) are followed from [14; Theorem 2.6]. ∎ Corollary 4: A semiprime right Goldie Ckring is commutative if its classical ring of quotient satisfies any Copyright © 2012 SciRes. APM
N. M. MUTHANA, S. K. NAUMAN Copyright © 2012 SciRes. APM 219 one of the properties (P1)(P15) as listed in Lemma 2. Acknowledgements This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant no. 3043/429. The authors, therefore, ac knowledge with thanks DSR technical and financial support. REFERENCES [1] L. Chuang and J. S. Lin, “On Conjecture of Herstein,” Journal of Algebra, Vol. 126, No. 1, 1989, pp. 119138. doi:10.1016/00218693(89)903220 [2] A. Klein, I. Nada and H. E. Bell, “Some Commutativity Results for Rings,” Bulletin of the Australian Mathe matical Society, Vol. 22, No. 2, 1980, pp. 285289. doi:10.1017/S0004972700006584 [3] J. Lambek, “On the Representation of Modules by Sheaves of Factor Modules,” Canadian Mathematical Bulletin, Vol. 14, 1971, pp. 359368. doi:10.4153/CMB19710651 [4] J. C. McConnell and R. Robson, “Noncommutative No etherian Rings,” AMS, 2001. [5] K. R. Goodearl, “Von Neumann Regular Rings,” Mono graphs and Studies in Math. 4 Pitman, 1979. [6] G. Marks, “On 2Primal Ore Extensions,” Communica tions in Algebra, Vol. 29, No. 5, 2001, pp. 21132123. doi:10.1081/AGB100002173 [7] S. U. Hwang, Y. C. Jeon and K. G. Park, “On NCI Rings,” Bulletin of the Korean Mathematical Society, Vol. 44, No. 2, 2007, pp. 215223. doi:10.4134/BKMS.2007.44.2.215 [8] G. F. Birkenmeier, H. E. Heatherly and E. K. Lee, “Com pletely Prime Ideals and Associated Radicals, Ring The ory (Granville, OH 1992),” World Sci. Publ. River H, Edge, 1993, pp. 102129. [9] H. E. Bell, “NearRings, in Which Every Element Is a Power of Itself,” Bulletin of the Australian Mathematical Society, Vol. 2, No. 3, 1970, pp. 363368. doi:10.1017/S0004972700042052 [10] K. Ham, Y. Jeon, J. Kang, N. Kim, W. Lee, Y. Lee, S. Ryu and H. Yang, “IFP Rings and NearIFP Rings,” Journal of the Korean Mathematical Society, Vol. 45, No. 3, 2008, pp. 727740. doi:10.4134/JKMS.2008.45.3.727 [11] H. K. Kim, N. K. Kim, M. S. Jeong, Y. Le e, S. J. Ry u a nd D. E. Yeo, “On Conditions Provided by Nil Radicals,” Journal of the Korean Mathematical Society, Vol. 46, 2009, pp. 10271040. [12] M. B. Rege and S. Chhawchharia, “Armendariz Rings,” Proceedings of the Japan Academy, Series A, Mathe matical Sciences, Vol. 73, No. 1, 1997, pp. 1417. [13] T.K. Lee and T.L. Wong, “On Armendariz Rings,” Houston Journal of Mathematics, Vol. 29, No. 3, 2003, pp. 583593. [14] Y. C. Jeon, H. K. Kim, Y. Lee and J. S. Yoon, “On Weak Armendariz Rings,” Bulletin of the Mathematical Society, Vol. 46, 2009, pp. 135136. [15] G. Marks, “A Taxonomy of 2Primal Rings,” Journal of Algebra, Vol. 266, No. 2, 2003, pp. 494520. doi:10.1016/S00218693(03)003016
