ON MAXIMAL IDEAL OF SKEW POLYNOMIAL RINGS OVER A DEDEKIND DOMAIN

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1 Far East Joural of Mathematcal Sceces (FJMS) Volume, Number, 013, Pages Avalable ole at Publshed by Pushpa Publshg House, Allahabad, INDIA ON MAXIMAL IDEAL OF SKEW POLYNOMIAL RINGS OVER A DEDEKIND DOMAIN Amr Kamal Amr Mathematcs Departmet Faculty of Mathematcs ad Natural Sceces Hasaudd Uversty Jl. Perts Kemerdekaa KM.10 Makassar, 9045, Idoesa e-mal: amrkamalamr@yahoo.com Abstract Let D be ay rg wth detty 1, σ be a edomorphsm of D, ad δ be a left σ -dervato. The skew polyomal rg over D a determate x, R = D[ x; σ, δ] cossts of polyomals a x + 1 a 1x + + a0, where a D wth stadard coeffcet-wse addto ad multplcato rule xa = σ( a) x + δ( a) for all a D. Ths work vestgates the maxmal deal of D [ x; σ], where D s a Dedekd doma ad σ s a automorphsm of D. 1. Itroducto Ths paper studes maxmal deals of a skew polyomal rg over a Dedekd doma. Skew polyomal rgs are wdely used as the uderlyg rgs of varous lear systems vestgated the area algebrac system 013 Pushpa Publshg House 010 Mathematcs Subject Classfcato: 16S36. Keywords ad phrases: automorphsm, commutatve, deal, maxmal, skew polyomal. Submtted by K. K. Azad Receved March, 013

2 Amr Kamal Amr theory. These systems may represet mathematcal models comg from mathematcal physcs, appled mathematcs ad egeerg sceces whch ca be descrbed by meas of systems of ordary or partal dfferetal equatos, dfferece equatos, dfferetal tme-delay equatos, etc. If these systems are lear, the they ca be defed by meas of matrces wth etres o-commutatve algebras of fuctoal operators such as the rg of dfferetal operators, shft operators, tme-delay operators, etc. A mportat class of such algebras s called skew polyomal rg. The structure of deals of varous kd of skew polyomal rgs have bee vestgated durg the last few years. I [1], [], [3], [6], ad [7], prme deals of skew polyomal rg automorphsm type over Dedekd doma were cosdered. Ths paper vestgates the maxmal deals of a skew polyomal rg over a Dedekd doma.. Deftos ad Notatos We recall some deftos, otatos, ad more or less well kow ecessary facts. I the frst part, we recall some facts about Dedekd doma. Some characterstcs of Dedekd doma assocated wth ts deals preseted the followg theorem. Theorem.1 (Hugerford [4]). The followg codtos o a tegral doma D are equvalet: 1. D s a Dedekd doma,. every proper deal D s uquely a product of a fte umber of prme deals, 3. every ozero deal D s vertble. By Theorem.1, every proper deal D s uquely a product of a fte umber of prme deals. Based o ths statemet, obtaed oe type of relatoshp betwee the prme deals, as stated the followg lemma. Lemma.1 (Osserma [8]). Let P...,, P1, P, P be prme deals of a Dedekd doma. If P P1 P P, the P = P for some.

3 O Maxmal Ideal of Skew Polyomal Rgs 3 I the secod part, we recall some deftos, otatos ad more or less well kow ecessary facts about skew polyomal rg. Defto.1 (McCoel ad Robso [7]). Let D be a rg wth detty 1, σ be a edomorphsm o the rg D, ad δ be a σ -dervatve o the rg D. The skew polyomal rg over D wth respect to the skew dervato ( σ, δ) s the rg cosstg of all polyomals over D wth a determate x deoted by: σ 0 D[ x;, δ] = { f ( x) = a x + + a a D} satsfyg the followg equato, for all a D, xa = σ( a) x + δ( a). The otatos D [ x; σ] ad D [ x; δ] stad for the partcular skew polyomal rg where respectvely δ = 0 ad σ s the detty map. Oe mportat role the vestgato of the structure of a rg s the detfcato of ts deals. Ths paper vestgated maxmal deals of the skew polyomal rg D [ x; σ]. The vestgato wll be doe by explotg the kowledge of the maxmal deal of the rg D. I preparato for our aalyss of the type of deals occurred whe prme deals of a skew polyomal rg D [ x; σ] are to the coeffcet rg D, we cosder σ -deal, δ -deal, ( σ, δ) -deal, σ -prme deal, δ -prme, ad ( σ, δ) -prme deals of D. Defto. (Goodearl [3]). Let Σ be a set of maps from the rg D to tself. A Σ -deal of D s ay deal I of D such that α ( I ) I for all α Σ. A Σ -prme deal s ay proper Σ -deal I such that wheever J, K are Σ -deal satsfyg JK I, the ether J I or K I. I the cotext of a rg D equpped wth a skew dervato ( σ, δ), we shall make use of the above defto the cases Σ = { σ}, Σ = { δ} ad Σ = { σ, δ}; ad smplfy the prefx Σ to, respectvely, σ, δ, or ( σ, δ). Accordg to the above defto, we ca coclude that f I s a prme

4 4 Amr Kamal Amr deal ad also σ -deal, the I s a σ -prme deal. The relatos betwee prme deal wth σ -deal are gve the followg two lemmas. Lemma. (Goodearl [3]). Let σ be a automorphsm o R ad I be a σ -deal of R. If R s a Noethera rg, the σ ( I ) = I. Lemma.3 (Goodearl [3]). Let σ be a automorphsm o Noethera rg R ad I be a σ -deal of R. The I s a σ -prme deal f ad oly f there + 1 exsts a prme deal P cosstg I ad postf teger such that σ ( P) = P ad I = P σ( P) σ ( P). 3. The Ma Results Lemma 3.1. Let σ be a automorphsm o Dedekd doma D ad p be a deal whch s ot a prme deal but σ -deal of D. The there exsts a prme deal m of D such that m p ad m = m + p. Proof. Accordg to the codtos o the lemma ad usg Lemma.3, the there exsts prme deal m cosstg p ad postf teger such that ( m ) = m ad ( ) p = m σ m σ ( m ). Ths leads to σ +1 m m + p p mα( m) σ ( m). Moreover, from Lemma.1 we kow that the set of prme deals cosstg p s { m, σ ( m),..., σ ( m)}. Assume that m m + p, the usg Theorem.1, we get m m + p = mσ 1( m) σ k ( m) for some,..., k { 1,..., }. Moreover, 1 m m + p = mσ 1 ( m) σ k ( m) k σ m σ 1 ( m) σ ( m) 1( m ).

5 O Maxmal Ideal of Skew Polyomal Rgs 5 Sce m s a maxmal deal, m = σ 1 ( m). Ths cotradcts wth p = m σ( m) σ ( m) ad p m. Theorem 3.1. Let R = D[ x; σ], where D s a Dedekd doma ad σ s a automorphsm. Let P be a mmal prme deal of R, where P = p [ x; σ] ad p s a σ -prme but ot prme deal of D. If m s a maxmal deal cosstg p where σ ( m) m, the M = m + xr s maxmal deal of R ad M = M + P. Proof. Let N be a deal of R ad M N. The there exsts a ( x) = 1 ax + a 1x + + a1x + a0 N but a( x) M. Sce ax 1 + a 1x + + a1x M N, the 0 a 0 N D ad a 0 m. Sce m s a maxmal deal, N D = D. Ths mples N = R. To show that M = M + P, t s eough to show that M M + P, because M M + P. Let f ( x) M = m + xr. The we ca wrte f ( x) the form as follows g0 f ( x) = a + x[ g x + g x + ] + 1 = 1 a + σ( g ) x + σ( g ) x + + σ( g ) x, 0 where 1 a m ad gx + g 1x + + g0 R. O the other had, Let M = ( m + xr)( m + xr) = m + xrm + mxr + xrxr. u x +1 xrm ad v x +1 mxr for = 1,...,.

6 6 Amr Kamal Amr The we ca choose w x +1 xrxr such that σ( g ) x = u x + v x + w x. 1 Therefore σ( g ) x M 0 M σ ( g ) x. + for = 1,...,. The ext, we wll show that We dvde the proof to two cases, amely: σ( g 0 ) m ad σ( g ). 0 m If σ( g ), the σ( g ) x m xr. O the other sde, f 0 m 0 M σ( g 0) m, the choose b m such that σ( b ) m. We ca choose such b because σ ( m) m. Now, we have the followg codtos: σ( g ), 0 m σ( b ) m, D m s a feld. So, we ca choose σ ( c) D\ m such that σ( g 0 ) = σ( c) σ( b) + l for some l m, ths mples 0 M σ( g ) x = σ( c) σ( b) x + lx = xcb + lx xrm + mxr. Furthermore, usg detty m = m + p Lemma 3.1, we get a M + P. Therefore, we have f ( x) M + P. Ths proves that M M + P. 4. Cocluso Let p be a σ -deal but ot a prme deal of a Dedekd doma D. The we ca choose a prme deal m of D such that m p ad m = m + p. Furthermore the prme deal m ca be exteded to be a maxmal deal of

7 O Maxmal Ideal of Skew Polyomal Rgs 7 skew polyomal rg R = D[ x; σ]. I ths case, we get that M = m + xr s a maxmal deal of R = D[ x; σ]. Ackowledgemet The author s very grateful to Professor Marubayash, Professor Pudj Astut, Professor Irawat, ad Doctor Ita Muchtad-Alamsyah for ther dscusso. Refereces [1] A. K. Amr, P. Astut ad I. Muchtad-Alamsyah, Mmal prme deals of Ore over commutatve Dedekd doma, JP J. Algebra Number Theory Appl. 16() (010), [] A. K. Amr, H. Marubayash, P. Astut ad I. Muchtad-Alamsyah, Corrgedum to mmal prme deals of Ore exteso over commutatve Dedekd doma ad ts applcato, JP J. Algebra Number Theory Appl. 1(1) (011), [3] K. R. Goodearl, Prme deals skew polyomal rgs ad quatzed Weyl algebras, J. Algebra 150 (199), [4] T. W. Hugeford, Algebra, Sprger-Verlag, New York, [5] R. S. Irvg, Prme deals of Ore exteso over commutatve rgs, J. Algebra 56 (1979), [6] R. S. Irvg, Prme deals of Ore exteso over commutatve rgs, II, J. Algebra 58 (1979), [7] J. C. McCoell ad J. C. Robso, Nocommutatve Noethera Rgs, Wley- Iterscece, New York, [8] B. Osserma, Algebrac Number Theory, Lecture Note, Dept. of Mathematcs, Uversty Calfora, 008.

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