International Journal of Algebra, Vol, 207, no 3, 27-35 HIKARI Ltd, wwwm-hikaricom http://doiorg/02988/ija2076750 On the Unit Group of a Cla of Total Quotient Ring of Characteritic p k with k 3 Wanambii A Wekea, Owino M Oduor 2, S Aywa 3 and Ojiema M Onyango 4,4 Department of Mathematic Mainde Muliro Univerity of Science and Technology PO Box 90-5000, Kakamega, Kenya 2 Department of Mathematic and Computer Science Univerity of Kabianga PO Box 2030-20200, Kericho, Kenya 3 Department of Mathematic Kibabii Univerity PO Box 536-50200, Bungoma, Kenya Copyright c 207 Wanambii A Wekea et al Thi article i ditributed under the Creative Common Attribution Licene, which permit unretricted ue, ditribution, and reproduction in any medium, provided the original work i properly cited Abtract Let R be a commutative completely primary finite ring The tructure of the group of unit for certain clae of R have been determined It i well known that completely primary finite ring play a crucial role in the endeavor toward the claification of finite ring Let G be an arbitrary finite group The claification of all finite ring R i o that U(R i ) G i till an open problem In thi paper, we conider S R to be a aturated multiplicative ubet of R and contruct a total quotient ring R S whoe group of unit i characterized, when char R S p k, k 3 It i oberved that U(R) U(R S ), ince R R S The cae when char R S p k, k, 2 have been tudied in a related work Mathematic Subject Claification: Primary 20K30; Secondary 6P0 Keyword: Total Quotient Ring, Unit Group, Localization, Completely Primary Finite Ring
28 Wanambii A Wekea et al Introduction Let R be a commutative completely primary finite ring with identity A ubet S R i multiplicative if S, 0 / S and xy S if x, y S Moreover, S i aturated if y / S implie x / S and y x The et of equivalence clae R S under the equivalence relation (r, ) (r, ) if and only if there exit v S uch that v(r r ) 0 hall be denoted by R S or S R The equivalence cla of (r, ) i denoted by [ r Uing addition and multiplication of quotient, it i eaily verified that S R {[ r r R, S i a commutative ring with identity [ We may alo regard S R a an R algebra with the tructure morphim f : R S R defined by r [ r verifiable a a ring homomorphim The R algebra tructure on S R i defined by r [ [ r rr Let R be a commutative ring with identity and S R be a aturated multiplicative ubet of R coniting of all the unit, then R S {[ r r R, S endowed with uual addition and multiplication of fraction i a total quotient ring Throughout thi paper, unle tated otherwie, R S S R denote the total quotient ring, U(R S ) denote the unit group of the total quotient ring R S, J(R S ) i the Jacobon radical of R S, GF (p r ) R S /J(R S ) The remaining notation are tandard and can be obtained from the reference eg [2 and [8 The characterization of finite ring into well known tructure i not complete In particular, Ayoub[ ha tudied the propertie of finite primary ring and their group of unit, Chikunji [2, 3 ha demontrated an in depth tudy on the tructure theory of completely primary finite ring He ha given the tructure of the unit group of certain clae of the aid finite ring and a review of ome other propertie of the ring Owino[7 ha tudied the unit of commutative finite ring with zero divior atifying the idempotent property It can be etablihed from thee finding that thee unit group are expreible a direct product of a cyclic group Z p r and the abelian p group + J(R) of order p (k )r where k, r are poitive integer Thi paper therefore eek to characterize U(R S ) when char p k, k 3 2 The Contruction of the total quotient ring of characteritic p k with k 3 Let R R U be an additive abelian group with identity, where R GR(p kr, p k ) i a Galoi ring of order p kr and characteritic p k and U be a
On the unit group of a cla of total quotient ring 29 h dimenional R module On thi group, define multiplication by the following relation: (i) When k, 2 pu i u i u j u j u i 0, u i r (r ) σ i u i (ii) When k 3, then p k u i p 2 γ ij u k i u j 0 for ome γ i,j R : i j where r R, i, j h, p i a prime integer, k and r are poitive integer and σ i i the automorphim aociated with u i Further, let the generator {u i for U atify the additional condition that, if u i U, then, pu i u i u j 0 From the given multiplication in R, we ee that if r + h i λ iu i and t + h i γ iu i, r, t R, λ i, γ i F are element of R, then, (r + h λ i u i )(t + i h γ i u i ) r t + i h {(r + pr )γ i + λ i (t + pr ) σ i u i It ha been verified that the given multiplication turn the additive abelian group R, into a commutative ring with identity (, 0,, 0) or jut if and only if σ i i the identity on R (cf ee [6, 8) Let J(R) be it Jacobon radical and S U(R) be a multiplicatively cloed ubet of R The localization of R with repect to S yield[ a total quotient ring R S uch that every element [(r, ) R S i of the form (r + h i λ iu i, α + pr + h i λ iu i ) where λ i F p, pr + h i λ iu i J(R) and α S Let [(r, ), [(r 2, 2 ) R S Define multiplication on R S a follow: i [(r, )[(r 2, 2 ) [(r r 2, 2 ) ( ) with the uual addition of fraction on R S, the multiplication defined by ( ) turn R S into a finite local ring with identity For any prime integer p and poitive integer k, h and r the contruction yield a total quotient ring in which (J(R S )) k [(0, ) and (J(R S )) k [(0, ) Thee ring are completely primary and therefore we can employ well known procedure to tudy their unit group 3 Preliminary Reult The following Theorem, Lemmata and Propoition hall be ued in the equel Theorem (cf [2) Let G be a cyclic finite group uch that G n, then G Z n Theorem 2 (cf [5) Let R be a finite ring Then every left unit i a right unit and every left zero divior i a right zero divior Furthermore, every element of R i either a zero divior or a unit
30 Wanambii A Wekea et al Theorem 3 (Property of localization of Completely primary ring) The only maximal ideal of S R are S J(R) where J(R) i a maximal ideal of R uch that J(R) S φ Lemma (cf [8) Let R S be a total quotient [ ring of the contruction in ection 2, then U(R S ) i cyclic if and only if + J(R S ) i cyclic Moreover [ U(R S ) < ξ > + J(R S ) a direct product of the p group < ξ > of order p r [ + J(R S ) by the cyclic ubgroup Lemma 2 (cf [8) Let ann(j(r S )) be the two ided annihilator of J(R S ) then + ann(j(r S )) i a ubgroup of + J(R S ) [ [ Theorem 4 Let R be a commutative ring with identity Suppoe S R i a multiplicative, then there exit a prime and maximal ideal P with repect to incluion among all the ideal in the complement of S in R Theorem 5 Let R be a commutative ring with identity Then the following are equivalent: i S i aturated ii R S i Λ P i where {P i Spec(R) uch that P i S φ Theorem 6 (Univeral property of the Quotient Ring) The canonical homomorphim f : R S R defined by f(r) [ r i an R algebra homomorphim, o that f atifie the following axiom i f i a ring homomorphim and f() U(S R) ii If θ : R R i a ring homomorphim with θ(s) U(R ), then there exit a unique homomorphim ψ : S R R uch that ψ f θ Theorem 7 R Then Let S be a multiplicative et of a unital commutative ring i Proper ideal of the ring S R are of the form IS R S I { i with I R an ideal and I S φ i I, S ii Prime ideal in S R are of the form S P where P i prime in R and P S φ
On the unit group of a cla of total quotient ring 3 Theorem 8 Let R be a commutative ring with identity and S R a multiplicative ubet, I i an ideal of R and S Im(S), the natural image of S in R/I, then S R/S I S (R/I) a an R algebra Remark In order to completely claify[ the group of unit of the ring contructed, we determine the tructure of + J(R S ) Since U(R S ) i [ abelian +J(R S ) i a normal ubgroup of U(R S ) Particularly, Let R S be the total quotient ring of the contruction ection 2, then +J(R S ) [ i an abelian p ubgroup of [ the unit group U(R S ) For brevity we hall in the equel denote the identity a [ 4 Main reult Propoition (cf [7) Let R S be the total quotient ring contructed above and J(R S ) be it i Jacobon radical, then {[ (i) J(R S ) (pr + h i λ iu i, α + pr + h i λ iu i ) r R, λ i Z p, u i R, α U(R) (ii) (J(R S )) k {[ (p k r, α + p k r ) r R, α U(R) and (iii) (J(R S )) k [(0, ) Let R S be the total quotient ring of the contruction in Section 2 with maximal ideal J(R S ) uch that (J(R S )) k [(0, ) and (J(R S )) k [(0, ) Then R S i of order p (k+h)r Since R S i of the given order and U(R S ) R S J(R S ), then U(R S ) p (k )r+rh (p r ) and + J(R S) p (k )r+rh So + J(R S) i an abelian p-group Propoition 2 Let R S be a local ring of characteritic p k with k 3 Then the group of unit U(R S ) of R S contain a cyclic ubgroup < ξ > of order p r and U(R S ) i a emi direct product of [ + J(RS ) by < ξ > In order to completely claify the group of unit of the total quotient ring, we determine the tructure of [ + J(RS ) We now give the generalization a follow; Propoition 3 The tructure of the unit group U(R S ) of the total quotient ring contructed in ection 2 with characteritic p k, k 3, r and h i a follow: [ + J(R S ) { Z2 Z 2 k 2 Z r 2 k (Z r 2) h if p 2; Z r p k (Z r p) h if p 2
32 Wanambii A Wekea et al and U(R S ) { Z2 r Z 2 Z 2 k 2 Z r 2 k (Z r 2) h if p 2; Z p r Z r p k (Z r p) h if p 2 Proof Let λ,, λ r R with λ uch that λ,, λ r R /pr form a bai for R /pr regarded a a vector pace over it prime ubfield F p Let,, h S U(R) Now U(R S ) U (R S /J(R S )) ([ ) + J(R S ) ([ ) Zp r + J(R S ) Since the two cae do not overlap, we conider them eparately: Let p 2 The tructure of [ + J(RS ) are obtained a follow; Suppoe l,, r, i h and let ψ R uch that x 2 + x + ψ 0 over R /pr ha no olution in the field R /pr and ψ R /pr, we obtain the following reult: Alo for l 2,, r, ([ + λ ) 2 lu ([ ) 2 [ + 2k λ ([ + 4ψ ) 2 k 2 [ ([ + 2λ ) 2 k l [ ([, + λ ) 2 lu 2, [ [ ([,, + λ ) 2 lu h [ for every l,, r Now, conider poitive integer α, β, κ l, τ l,, τ hl with α 2, β 2 k 2, κ l 2 k, τ il p( i h, l r), we notice that the equation ([ ) α ([ + 2k λ + 4ψ l {([ + λ ) τl lu l ) β l l2 {([ + λ ) τhl lu h {([ + 2λ ) κl l {([ + λ ) τ2l lu 2 {[
On the unit group of a cla of total quotient ring 33 will imply α 2, β 2 k 2, κ l 2 k for l 2,, r and τ il 2 for every l,, r and i h If we et {([ ) α H + 2k λ α, 2 {([ ) α G β,, 2 k 2 T l + 4ψ {([ + 2λ ) κ l κ,, 2 k ; l 2,, r {([ S l + λ ) τ lu τ, 2 {([ S 2l + λ ) τ2 lu 2 τ 2, 2 S hl {([ + λ ) τh lu h τ h, 2 ; it i eaily noticed that H, G, T l, S l, S 2l,, S hl are cyclic ubgroup of the group +J(R S) and they are of the order indicated in their definition Since, [ [ + 2k λ > + 4ψ < l [ + λ lu > l [ + λ lu h > l < l2 [ + 2λ l > [ + λ lu 2 > > p (h+k )r, the [ interection of any pair of the cyclic ubgroup yield the identity group, and the product of the h + 3 ubgroup H, G, Tl, S l, S 2l,, S hl i direct and exhaut the group + J(R S) Let p 2 For l,, r, ([ ) p k + pλ l [ ([, + λ ) p u [ ([,, + λ ) p hu h [
34 Wanambii A Wekea et al For poitive integer α l, β l,, β hl with α l p k, β il p( i h), we notice that the equation {([ ) αl { ([ + pλ l + λ ) βl u l { ([ l l + λ hu h ) βhl {[ will imply α l p k, β il p( i h, l,, r) If we et {([ ) α T l + pλ l α,, p k S l {([ + λ ) α u α,, p k {([ S hl + λ ) α hu h α,, p k We ee that T l, S l,, S hl are all cyclic ubgroup of [ +J(RS ) and they are of the order indicated by their definition Since l [ + pλ l > l [ + λ u > l [ + λ hu h > p (h+k )r and the interection of any pair of the cyclic ubgroup give the identity group, the product of the (h + )r ubgroup T l, S l,, S hl i direct and the product exhaut the group [ + J(RS ) Acknowledgement My thank to Dr Owino Maurice for hi inightful comment and direction May God ble you A lot of thank to Mainde Muliro Univerity of Science and Technology for their financial upport toward the reearch Reference [ C W Ayoub, On finite primary ring and their group of unit, Compoitio Math, 50 (969), 247-252 [2 C J Chikunji, On unit group of completely primary finite ring, Mathematical Journal of Okayama Univerity, 50 (2008), 49-60
On the unit group of a cla of total quotient ring 35 [3 C J Chikunji, A review of the theory of completely primary finite ring, Conference Proceeding of the 2nd Strathmore International Mathematic Conference (SIMC 203), 2-6 Augut 203, Strathmore Univerity, Kenya [4 F Kainrath, A note on quotient formed by unit group of emi local ring, Houton Journal of Mathematic, 2 (998), no 4, 63-68 [5 N O Omolo, MO Owino, On the tructure of quotient group, International Journal of Pure and Applied Mathematic, 54 (2009), no 4, 497-502 [6 M O Owino, E Mmai and M Ojiema, On the quotient group of ubgroup of the unit of a cla of completely primary finite ring, Pure Mathematical Science, 4 (205), no 3, 2-26 http://doiorg/02988/pm205556 [7 M O Owino and CJ Chikunji, Unit Group of a Certain cla of Commutative Finite Ring, Journal Math Sci, 20 (2009), no 3, 275-280 [8 M O Owino, M O Ojiema and E Mmai, Unit of commutative completely primary finite ring of characteritic p n, International Journal of Algebra, 7 (203), no 6, 259-266 http://doiorg/02988/ija2033026 Received: Augut, 206; Publihed: April 24, 207