A NOTE ON WEAKLY VON NEUMANN REGULAR POLYNOMIAL NEAR RINGS

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1 IJMS, Vol. 11, No. 3-4, (July-December 2012), pp Serials Publicatios ISSN: X A NOTE ON WEAKLY VON NEUMANN REGULAR POLYNOMIAL NEAR RINGS P. Jyothi & T. V. Pradeep Kumar Abstract: The mai aim of this paper is to ivestigate the fudametal structures of Regular polyomial ear rigs, the cocept of degree, least degree ad symbolically zero polyomials Mathematics Subject Classificatio: 16Y30. Keywords: Near-rig, Regular ear-rig, Polyomial ear rig, Regular polyomial ear rig. 1. INTRODUCTION First we recall some defiitios ad results related to Near rigs Polyomial ear-rigs, Regular ear-rigs over a right ear rig with idetity. Proofs other iformatio could be foud i [3], [5]. Defiitio 1.1: A oempty set N is said to be a Right ear-rig with two biary operatios + ad. If (i) (N, +) is a group (ot ecessarily abelia) (ii) (N,.) is a semi group ad (iii) (x + y) z = xz + yz for all x, y, z N. Example 1.2: Let Z be the set of positive, egative itegers with 0, the (z, +,.) is a ear rig with usual additio ad multiplicatio. Defiitio 1.3: A ear rig N is said to be Regular ear rig if for each elemet x N the there exists a elemet y N such that x = xyx. Example 1.4: (Regular ear rig) (i) M ( ) ad M 0 ( ) are regular ear rigs (Beidlema (3)) (ii) Costat ear rigs (iii) Direct sum ad product of ear fields. 1 st Iteratioal Coferece o Mathematics ad Mathematical Scieces (ICMMS), 7 July 2012.

2 374 P. Jyothi & T. V. Pradeep Kumar Note 1.5: Homomorphic images, direct sums ad direct products of regular ear-rigs are regular. 1.6: A ormal subgroup I of a ear rig N is said to be (i) A right ideal if IN I (ii) A left ideal if x ( + y) xy I for all x, y N, I (iii) Ideal if it is both left ad right ideal. Result 1.7: (Beidlema (3)) Let N 1. N is regular iff N such that e = e q N; N := Ne. Result 1.8: (Beidlema (3)) : A regular ear rig with idetity cotais o ozero il N-sub groups. : (Beidlema (3)). Defiitio 1.9: A regular ear rig N is called strogly regular if {0} N 1 ad if N fulfills the coditios that (i) N = N 0 has o o zero ilpotet elemets. (ii) All idempotets of N are cetral. (iii) N is a sub direct product of ear-field. Theorem 1.10: (Mari (1)) N 0 1 is strogly regular if N x N such that = x WEAKLY VON NEUMANN REGULAR NEAR RINGS All ear rigs cosidered i this paper are assumed to be zero symmetric ear rigs, ad have idetity elemet 0; all modules are uital. A ear rig N is reduced if its il radical is zero. The followig statemets o a ear rig N are equivalet: 1. Every fiitely geerated ideal of N is pricipal ad is geerated by a idempotet. 2. For each x i N, there is some y i N such that x = x 2 y. 3. Nis reduced 0-dimesioal ear rig. Defiitio 2.1: A ear rig N is called a weak vo Neuma regular ear rig (WVNR for short) if for every fiitely geerated ideals I ad J of N satisfyig I J N whe J is geerated by a idempotet elemet of N, the so is I. I particular, ay vo Neuma regular ear rig is a weak vo Neuma regular ear rig. Now, we give a class of a weak vo Neuma regular ear rig.

3 A Note o Weakly Vo Neuma Regular Polyomial Near Rigs 375 Example 2.2: If N is a ear rig i which the oly idempotet elemets are 0 ad 1, the N is a WVNR rig. I particular if N be a itegral domai or a local rig, the N is a WVNR ear rig. Theorem 2.3: The followig coditios o a ear rig N are equivalet: 1. N is a WVNR ear rig. 2. For each a Ne where e is a o uit idempotet elemet of N, the a Na 2 3. For each a Ne where e is a o uit idempotet elemet of N, the Na is direct summad of N. Proof: (1) (3): Let a N ad 1 e be a idempotet elemet of N such that a Ne. Sice e is o uit we have the cotaimets Na Ne ( ) N. From the defiitio of a WVNR ear rig, we ca write Na = Nf for some idempotet f N. It follows that Na N (1 f ) = N. (3) (2): Let a Ne where e is a o uit idempotet elemet of N ad let I be a ideal of N such that I Na = N. We ca write 1 = u + v for some u I ad v Ra. Multiplyig the above equality by u (resp., v) we get that u 2 = u (resp; v 2 = v). Thus I = Nu ad Na = Nv, therefore a = au + av = av = a 2 x, for some x N. (2) (1): Let J be a pricipal ideal geerated by a o uit idempotet elemet e of N, ad let I be a fiitely geerated ideal of N cotaied i J. It suffices to prove that if I = (a, b), the there exists a idempotet f i N such that I = Nf. Sice a J = Ne the a Na 2, also b Nb 2. Let u = ax ad v = by, where a 2 x = a ad b 2 y = b. Hece u ad v are idempotet elemets of N. The elemet f = u + v uv has the required property. The followig Corollary is a immediate cosequece of Theorem 2.3. Corollary 2.4: Let N be a ear rig. The the followig statemets are equivalet. 1. N is a vo Neuma regular ear rig. 2. N is a WVNR ear rig ad for every o uit elemet a of N there exists a Idempotet e 1of N such that a Ne. Now we give a ecessary ad sufficiet coditio for a direct product of rigs to be a WVNR ear rig. Theorem 2.5: Let (N i ) 1 i be a family of ear rigs, with 2. The the followig statemets are equivalet:

4 376 P. Jyothi & T. V. Pradeep Kumar Ni is a vo Neuma regular ear rig. i 1 Ni is a weak va Neuma regular ear rig. i 1 3. For each i {1, 2,, }, R i is a vo Neuma regular ear rig. Proof: Straight forward. Now we give a example of a o-wvnr Noetheria regular ear rig. Example 2.6: Let be a positive iteger such that 2, ad let Z be the rig of itegers. The z is ot a WVNR ear rig sice Z is ot a vo Neuma regular ear rig. Cosequetly, z is a o vo Neuma regular Noetheria ear rig. For a ideal I of a WVNR ear rig N/I is ot ecessarily a WVNR ear rig. For this we claim that Z/12Z is ot a WVNR ear rig, where Z is the rig of itegers. Ideed, 9 is a idempotet ad 9.2 = 6 but 6 2 = 0.Thus x for each x Z/12Z. By applyig coditio (2) of Theorem 2.4, we get the result. I the ext theorem, we give sufficiet coditio for N/I to be a WVNR ear rig. Theorem 2.7: Let N be a Regular ear rig ad let I be a primary ideal. The N/I is a WVNR ear rig. Proof: We deote a = a + I for every a N. To prove this theorem it is eough to show that N/I has exactly two idempotet elemets which are 0 ad 1. Let a N such that a is a o zero idempotet elemet of N/I. We have a 2 a I. Sice I is a primary ideal of N ad a I, there exists a o egative iteger such that (a 1) I. By the biomial theorem k k ( a 1) ( 1) a I. k 0 k We put a 2 = a + x. By iductio we claim that for each k 2, a k = a + x (1 + a + a a k 1 ). Ideed, it is certaily true for k = 2. Suppose the statemet is true for k, the we get the followig iequalities a k + 1 = a 2 + x (a + a a k 1 ) = a + x (1 + a a k 1 ). We coclude that for each oegative iteger, there is some x k I such that a k = a + x k. We ca also deduce that k ( 1 ) 1 ( 1) ( a xk ) I k 1 k.

5 A Note o Weakly Vo Neuma Regular Polyomial Near Rigs 377 k But ( 1 ) 1 ( 1) a ( 1) (1 a ), hece 1 a I ad so a = 1. By applyig k 1 k Example 2.2 we get N is a WVNR ear rig. REFERENCES [1] Bagley S., (1993), Polyomial Near Rigs, Distributor ad J. Ideals of Geeralized Cetralizer Near-Rigs, Doctoral Dissertatio, Texa A&M Uiversity. [2] Bagley S. W., (1997), Polyomial Near-Rigs Polyomials with Coefficiets from a Near-Rig, I: Saad, Thomse, Rds. Near-Rigs, Near-Fields ad Loops, Netherlads: Kluwer Academic Publishers, [3] J. C. Beidlema, (1969), A Note o Regular Near-Rigs, J. Idia Math. Soc., 33: [4] Meldrum J. D. P., (1985), Near-Rigs ad Their Liks with Groups, Marshfield, M.A., Pitma. [5] Pilz G., (1983), Near-Rigs, Amsterdam, North-Hollad, America Elsevier. [6] S. Leigh, (1969), Boolea Near-Rigs ad Weak Commutativity, Bull Austral. Math Soc., l: [7] Y. V. Reddy, ad C. V. L. N. Murthy, (1984), O Strogly Regular Near-Rigs, Proc. Ediburgh Math.Soc., 27: P. Jyothi Asst. Prof. i Mathematics, Laqshya istitute of Techology ad Scieces, Khammam. jyothi.puligadda.77@gmail.com T. V. Pradeep Kumar Asst. Prof. i Mathematics, ANU College of Egieerig ad Techology, Acharya Nagarjua Uiversity, pradeeptv5@gmail.com