THE STRUCTURE OF A RING OF FORMAL SERIES AMS Subject Classification : 13J05, 13J10.

THE STRUCTURE OF A RING OF FORMAL SERIES GHIOCEL GROZA 1, AZEEM HAIDER 2 AND S. M. ALI KHAN 3 If K is a field, by means of a sequence S of elements of K is defined a K-algebra K S [[X]] of formal series called Newton interpolating series which generalize the formal power series. We study algebraic properties of this algebra, and in the case when S has a finite number of distinct elements we prove that it is isomorphic to a direct sum of a finite number of algebras which are either equal to K[[X]] or to factor rings of K[X]. 2000 AMS Subject Classification : 13J05, 13J10. Key words: formal series, semi-local ring. 1. Introduction The K-algebra K[[X]] of formal power series is a basic structure in commutative algebra with applications in algebraic geometry. Also formal power series are useful tools in combinatorics as generating functions or in number theory. In the interpolation theory particular Newton series, with coefficients in a field K, constructed by means of Newton interpolating polynomials is a powerful tool either in archimedean or in non-archimedean analysis. Thus for K = C the convergence of a family of Newton series is the subject of [6]. Also a proof of a well-known result of Lindemann on the transcendency of e γ, when γ is an algebraic number (see [8], Theorem 6, Ch. 2, Sec. 3) is based on a Newton series with complex coefficients. In the non-archimedean case the Mahler series which are convergent Newton series are used for representation of continuous functions on Z p (see for example [1] or [7]). Applications of these series to approximate solutions of boundary value problems for differential equations are presented in [3] and [4]. All these convergent series belong to some subalgebras of a K-algebra of formal series called (formal) 1 Department of Mathematics and Computer Science, Technical University of Civil Engineering of Bucharest, 124 Lacul Tei, RO-020396, Bucharest, Romania, email:grozag@mail.utcb.ro 2 Department of Mathematics, Faculty of Sciences, Jazan University, Jazan, Kingdom of Saudi Arabia, email:azeemhaider@gmail.com 3 University of Education, Lahore, Pakistan, email:mohib.ali@gmail.com. 1

2 GHIOCEL GROZA, AZEEM HAIDER AND S. M. ALI KHAN Newton interpolating series defined below. So the structure of this algebra becomes an important goal for problems of previous types. Let K be a field and S = {α n } n 1 a fixed sequence of elements of K. We consider the polynomials i (1) u 0 = 1, u i = (X α j ), i 1 and the set of formal sums (2) K S [[X]] = {f = j=1 a i u i a i K}, two such expressions being regarded as equal if and only if they have the same coefficients. We call an element f from K S [[X]] a (formal) Newton interpolating series with coefficients in K defined by the sequence S. If K S [[X]] is endowed with a suitable addition and multiplication it becomes a K- algebra (see Section 2) which for α k = 0, for all k, is equal to K[[X]]. The main goal of this paper is to give a structure theorem of K S [[X]] for the case when the sequence S has a finite number of distinct elements. In order to prepare the proof of the main result, in Section 3, Theorem 1 we describe the units of K S [[X]], we prove an Isomorphism Theorem (Theorem 2) and we find the spectrum of maximal ideals of K S [[X]] (see Theorem 3). In Section 4 we prove (see Theorem 5) that it is isomorphic to a direct sum of a finite number of algebras which are either equal to K[[X]] or to factor rings of K[X]. Moreover its algebraic structure is uniquely determined by a finite number of non-negative integers, so-called the invariants of the sequence S. 2. Basic notations and definitions The following lemma is needed to give a suitable form of well-known Newton interpolating polynomial. Lemma 1. Let K be a field and let S = {α n } n 1 be a sequence of elements of K. Then every polynomial P K[X] can be written uniquely in the form P = p a i u i with a i K, where p is the degree of P and every u i is defined by (1). Proof. If P = p b jx j, then we may uniquely write P = a p u p + Q q, where a p = b p, Q q K[X] such that deg Q q < p. Now the lemma follows by induction on p.

THE STRUCTURE OF A RING OF FORMAL SERIES 3 If u i, u j are given by (1), we obtain that for every k, max {i, j} k i + j there exist the elements d k (i, j) in K uniquely determined such that (3) u i u j = i+j k=max {i,j} d k (i, j)u k. The following relationships are easily checked: (4) d k (i, j) = d k (j, i), (5) d i+j (i, j) = 1. We define addition and multiplication of two elements f = a i u i, g = b i u i, K S [[X]] as follows (6) f + g = and (7) fg = with (8) c k = where (a i + b i )u i (α,β) I(k) c k u k k=0 d k (α, β)a α b β, (9) I(k) = {(α, β) N N max {α, β} k, α + β k} and d k (α, β) are given in (3). It is easily seen that with these definitions of addition and multiplication K S [[X]] becomes a commutative K-algebra which contains K[X]. If f = a i u i K S [[X]], the smallest index i for which the coefficient a i is different from zero will be called the order of f and will be denoted by o(f). We agree to attach the order + to the element 0 from K S [[X]]. Then it is easy to prove that for f, g K S [[X]] (10) o(f + g) min {o(f), o(g)}, (11) o(fg) max {o(f), o(g)}.

4 GHIOCEL GROZA, AZEEM HAIDER AND S. M. ALI KHAN If we fix a positive real number δ < 1 and define the norm f of an element f of K S [[X]] by the formula (12) f = δ o(f), K S [[X]] becomes a K ultrametric normed vector space, where the norm on K is trivial. Moreover for f, g K S [[X]] (13) fg min{ f, g } holds and K S [[X]] is a topological K-algebra. Consider a Cauchy sequence {f n = a i,nu i } n 0 of elements from K S [[X]]. Then f = a i,iu i K S [[X]] and f = lim n f n. Hence it follows that K S [[X]] is a complete K ultrametric normed vector space and it is a completion of K[X] with respect to the topology defined by the norm (12). This implies as in the classical case (see for example [9], Ch.VII, 1) that the distributive law holds also for infinite sums and in particular we obtain that (14) u j a i u i = a i u i u j For a fix f K S [[X]] we denote L f : K S [[X]] K S [[X]] the K-linear application defined by (15) L f (g) = fg Then, by (3) and (14) (16) L f (u i ) = fu i = f j,i u j where f j,i K are uniquely determined by f. Since L f (u i+1 ) = f j,i+1 u j = fu i (X α i+1 ) = f j,i u j (X α i+1 ) = j=i+1 f j,i (u j+1 + (α j+1 α i+1 )u j ) = j=i it follows that j=i j=i+1 j=i (f j 1,i + (α j+1 α i+1 )f j,i )u j (17) f j,i+1 = f j 1,i + (α j+1 α i+1 )f j,i, for j = i + 1, i + 2,.... If g = b j u j, then by (14) and (16) we obtain (18) fg = b j (fu j ) = b j ( f i,j u i ) = i=j ( j b i f j,i )u j.

THE STRUCTURE OF A RING OF FORMAL SERIES 5 For f = a j u j if we take i = 0 in (16), we get for every j (19) f j,0 = a j. Now for a fix j, by (17) we obtain that f j,j = f j 1,j 1 + (α j+1 α j )f j,j 1 = f j 2,j 2 + (α j α j 1 )f j 1,j 2 + (α j+1 α j )(f j 1,j 2 +(α j+1 α j 1 )f j,j 2 ) = f j 2,j 2 +(α j+1 α j 1 )f j 1,j 2 + (α j+1 α j )(α j+1 α j 1 )f j,j 2, and by recurrence for all k, 1 k j it follows that k k r (20) f j,j = f j r,j k (α j+1 α j k+s ). r=0 s=1 In particular for k = j we obtain j r j (21) f j,j = f j r,0 (α j+1 α s ) = where 0 s=1 r=0 (α j+1 α s ) = 1. s=1 Similarly for i, j such that 1 k i < j, j r=0 a j r j r (α j+1 α s ), f j,i = f j 1,i 1 + (α j+1 α i )f j,i 1 = f j 2,i 2 + (α j α i 1 )f j 1,i 2 +(α j+1 α i )(f j 1,i 2 + (α j+1 α i 1 )f j,i 2 ) = f j 2,i 2 + (α j + α j+1 α i 1 α i )f j 1,i 2 + (α j+1 α i )(α j+1 α i 1 )f j,i 2 and generally k (22) f j,i = f j r,i k P j r,i k, r=0 where P j r,i k are polynomials in α 1,..., α j+1 with integer coefficients and (23) P j,i k = (α j+1 α i )(α j+1 α i 1 )...(α j+1 α i k+1 ). For any sequence S = {α n } n 1 we define the set (24) I S = {i α i α j for all j < i} and for any γ K we put (25) I S (γ) = {i α i = γ}. We call the sequence S = {α n } n 1 purely periodic if I S is finite set having m elements and, for each positive integer i less or equal to m, α i = α i+jm, j = 1, 2,.... Now if S = {α k } k 1 is a sequence of elements of K such that I S s=1

6 GHIOCEL GROZA, AZEEM HAIDER AND S. M. ALI KHAN is a finite set, we say that S has canonical form if there exist the positive integers n 1... n s such that α 1 = α 2 =... = α n1 = γ 1, α n1 +1 =... = α n1 +n 2 = γ 2,..., α n1 +...+n s 1 +1 =... = α n1 +...+n s = γ s with γ i γ j for all i j and the subsequence T = {α k } k>n1 +...+n s is purely periodic sequence such that each α k γ i for all i = 1, 2,..., s. Let S be a sequence such that I S = n is finite. We may write I S = {i 1,..., i s, i s+1,..., i n } where I S (α ik ) = n k is finite for k s, with n 1... n s, and infinite for k > s. Then we call the numbers s, m = n s, n 1,..., n s, the invariants of the sequence S. 3. Algebraic properties of the ring K S [[X]] In order to describe the units of the ring K S [[X]] we need the following lemma which is an easy consequence of (21). Lemma 2. If α n+1 = α i for i < n then f n,n = f i 1,i 1. Theorem 1. An element f K S [[X]] is a unit if and only if f i 1,i 1 0 for all i I S. Proof. If f = a i u i K S [[X]] is a unit then by (18) there exists an element g = b j u j K S [[X]] such that { j b i f j,i }u j = 1. Hence b 0 f 0,0 = 1 and j b i f i,j = 0 for all j = 1, 2,... Suppose contrary that there exists t + 1 I S such that f t,t = 0. Since b 0 f 0,0 = 1 and ( j b i f j,i )(α t+1 α 1 )(α t+1 α 2 )...(α t+1 α j ) = 0 for all j = 1, 2,...t, we obtain b 0 f 0,0 + t {( j b i f j,i )(α t+1 α 1 )(α t+1 α 2 )...(α t+1 α j )} = 1, j=1 which implies (26) t j (α t+1 α k ){f j,j + b j where k=1 0 k=1 t i=j+1 f i,j (α t+1 α j+1 )...(α t+1 α i )} = 1, (α t+1 α k ) = 1. Now by replacing j with t and k with t j in (20) it follows that for j = 0, 1,..., t 1 f t,t = f j,j + α j+1 )(α t+1 α j+2 )...(α t+1 α i ) and by (26) f t,t t b j k=1 t i=j+1 f i,j (α t+1 j (α t+1 α k ) = 1.

THE STRUCTURE OF A RING OF FORMAL SERIES 7 Since f t,t = 0 we obtain a contradiction which implies that f i 1,i 1 0 for all i I S. Conversely, if f K S [[X]] and f i 1,i 1 0 for all i I S, we can find b 0 such that b 0 f 0,0 = 1. Since by Lemma 2 f n,n 0 for all n N. there j exists the elements b j K such that b i f i,j = 0. Hence g = b i u i verifies fg = 1. We study when two K-algebras K S [[X]] and K S [[X]] are isomorphic. Lemma 3. Let S = {α n } n 1 and S = {β n } n 1 be two sequences of elements of K and let u i and respectively v i be the associated polynomials defined in (1). If a ) for every i (27) u i = v i + (28) v i = u i + where δ j,i, γ j,i K and i 1 j=n i δ j,i v j = F i (v ni, v ni +1,..., v i ), i 1 j=m i γ j,i u j = G i (u mi, u mi +1,..., u i ), (29) F i (G ni, G ni +1,..., G i ) = u i, (30) G i (F mi, F mi +1,..., F i ) = v i ; b ) (31) lim i n i = and lim i m i =, then the map φ : K S [[X]] K S [[X]] defined by (32) φ(u i ) = F i (v ni, v ni +1,..., v i ) and for every f = a i u i K S [[X]] (33) φ(f) = is a K algebra isomorphism. a i φ(u i ) Proof. Define ψ : K S [[X]] K S [[X]] such that (34) ψ(v i ) = G i (u mi, u mi +1,...u i )

8 GHIOCEL GROZA, AZEEM HAIDER AND S. M. ALI KHAN and for every g = b i v i K S [[X]] (35) ψ(g) = b i ψ(v i ). Then by (31),(33) and (35) the maps φ and ψ are well defined and continuous maps with respect to the corresponding norms defined by (12). The relations (29) and (30) imply that the restricted mappings φ and ψ on K[X] are inverses. Since K[X] is dense in K S [[X]] and K S [[X]] we obtain that φ and ψ are inverses and hence φ is bijective map. Because φ is the identity map on K[X] it follows that φ is also a K algebra morphism. Hence we obtain that K S [[X]] and K S [[X]] are isomorphic K algebras. Theorem 2. Suppose S = {α k } k 1 is a sequence of elements of K and π : N N is a bijective map such that S = {β k } k 1, where β k = α π(k). Then K S [[X]] and K S [[X]] are isomorphic K algebras. Proof. Consider u i = i k=1 1 every u i can be written in the form i 1 (36) u i = v i + γ j,i v j, (X α k ) and v i = i (X β k ). By Lemma where γ 0,i = u i (β 1 ). Let t 1 be the smallest index i such that v 1 divides u i in K[X]. Then for every i t 1 u i (β 1 ) = 0 and by (36) we obtain that (37) u i = v i + i 1 j=c 1 γ j,i v j, where c 1 1 and γ c1,t 1 0. Consider q 1 c 1 the greatest index such that v q1 divides u t1 in K[X]. Then by (37), for all i t 1, (38) u i v q1 = v i v q1 + i 1 j=c 1 γ j,i v j v q1. Let t 2 be the smallest index i greater than t 1 such that v q1 +1 divides u t2. Then by (38) γ c1,t 2 = ut 2 v q1 (β q1 +1) = 0 and for all i t 2 (39) u i = v i + i 1 j=c 2 γ j,i v j, k=1

THE STRUCTURE OF A RING OF FORMAL SERIES 9 where γ c2,t 2 0 and c 2 > c 1. Thus by recurrence we obtain that for every i (40) u i = v i + i 1 j=n i γ j,i v j = F i, where γ j,i K and lim n i =. Similarly, since α k = β π 1 i (k), it follows that (41) v i = u i + i 1 j=m i δ j,i u j = G i, where δ j,i K and lim i m i =. By Lemma 1, (40) and (41) F i (G ni,..., G i ) = u i and G i (F mi,..., F i ) = v i. Thus the theorem follows from Lemma 3. Remark 1. Suppose that S = {α k } k 1 is a sequence such that I S is a finite set. Then there exists a bijective map π : N N such that S = {β k } k 1, where β k = α π(k) has canonical form. By Theorem 2 it follows that K S [[X]] and K S [[X]] are isomorphic K-algebras. Thus it is enough to study the algebraic properties of K S [[X]] in the case when S has canonical form. Lemma 4. Let α k be an element of S. Then g K S [[X]] belongs to the ideal generated by X α k if and only if g k 1,k 1 = 0. Proof. We denote by < X α k > the ideal generated by X α k. Then for an element f = a i u i of K S [[X]] (42) (X α k )f = a 0 (α 1 α k ) + (a i 1 + a i (α i+1 α k ))u i If g = b i u i is an element of < X α k >, then from (42) we can find the elements a i such that (43) b 0 = a 0 (α 1 α k ), b i = a i 1 + a i (α i+1 α k ), for i = 1, 2,.... By (43) and (21) it follows that k 1 g k 1,k 1 = r=0 b k 1 r k 1 r s=1 (α k α s ) = 0. Conversely if g = b i u i is an element of K S [[X]] such that g k 1,k 1 = 0, then by using (21), (43) and Lemma 2 we can find an element f =

10 GHIOCEL GROZA, AZEEM HAIDER AND S. M. ALI KHAN a i u i K S [[X]] such that g = (X α k )f and hence g < X α k >. Lemma 5. The ideal generated by X α i for each i I S is a maximal ideal in K S [[X]]. Proof. We consider an element h K S [[X]] such that h < X α k >. Then by Lemma 4 h k 1,k 1 0. Since for every i I S, i k the elements b i 1 in (43) can be chosen arbitrary, we can find an element g = b i u i < X α k > such that (g + h) i 1,i 1 0 for all i I S, i k. If i = k, by Lemma 4 (g + h) k 1,k 1 = g k 1,k 1 + h k 1,k 1 = 0 + h k 1,k 1 = h k 1,k 1 0. Hence by Theorem 1 g + h is unit in K S [[X]], which shows that < X α k > is a maximal ideal in K S [[X]]. Theorem 3. K S [[X]] is a semi-local ring if and only if I S is a finite set. Moreover in this case all maximal ideals are M i =< X α i >, where i I S. Proof. Suppose the contrary that K S [[X]] is a semi-local ring and I S is infinite. Then by Lemma 5 the ideal generated by X α i, for each i I S, is a maximal ideal in K S [[X]]. Hence there exist an infinite number of maximal ideals in K S [[X]] which is a contradiction. This implies that I S is a finite set. Conversely, suppose I S = {i 1,..., i n }. By Lemma 5 each ideal generated by X α ik, for each i k I S, is a maximal ideal in K S [[X]]. Let M be a maximal ideal of K S [[X]] different from all M ik =< X α ik >, with i k I S. Since M + M i1 M i2...m in = K S [[X]], we can find an element f M such that f < X α ik >, for each i k I S. Then by Lemma 4 f ik 1,i k 1 0 for all i k I S and by Theorem 1 f is unit in K S [[X]], a contradiction which shows that all maximal ideals in K S [[X]] are M ik with i k I S. Thus K S [[X]] is a semi-local ring. The following example shows that when I S is infinite K S [[X]] contains maximal ideals which are different from < X α k >, for every k. Example 1. Consider S = {α n } n 1 such that α i α j for every i j. Because u n (α k ) = 0 for n k, every f = a i u i K S [[X]] defines a map denoted also by f from S to K. Moreover, u k (α k+1 ) 0 implies that every f is uniquely determined by its values at α k, k N and K S [[X]] is isomorphic to the K-algebra of all the functions from S to

THE STRUCTURE OF A RING OF FORMAL SERIES 11 K. For every j N we consider h j K S [[X]] such that { 1, if k j (44) h j (α k ) = 0, if k > j. Then for every nonzero h I =< h 1,..., h n,... > there exist m, n N such that h(α m ) 0 and h(α k ) = 0 for every k n. Hence it follows that I K[X] = {0}, I < X α k > for every k and I is not finitely generated. Thus K S [[X]] contains maximal ideals which are different from < X α k >, for every k. 4. Structure of the ring K S [[X]] In this section we prove that the structure of the K-algebra K S [[X]] is uniquely determined by the invariants of S, when I S is a finite set. The following lemma describes an essential isomorphism of K- algebras. ( We note that canonic isomorphisms of K-vector spaces as n 1 ) f a i u i, a i u i /u n 1 are not K-algebra isomorphisms. Lemma 6. Let S = {α i } i 1 be a sequence of elements of K such that, for a fixed n N, α 1 = α 2 =... = α n = γ and α i γ, for all i > n. Then K S [[X]] = K[X]/ < X n > K S [[X]], where S = {β i } i 1, with β i = α n+i. Proof. By (18) it follows that an element f = e i u i is an idempotent if and only if j e i f j,i = e j for all j 0. In order to find an idempotent we consider f such that e 0 = 1, e 1 = e 2 =... = e n 1 = 0. Then by (17), for every i = 1, 2,..., n 1 and j = 0, 1,..., i 1, it follows that (45) f i,j = 0 and f i,i = 1. By successive applications of (17) and (19) we obtain that ( n n ) l f n+k,n = e k + (α k+1+ij γ) e k+l. If we put the conditions l=1 i 1,...,i l =l j=1 (46) f n+k,n = 0,

12 GHIOCEL GROZA, AZEEM HAIDER AND S. M. ALI KHAN for every k = 0, 1,... we find (47) ( ( 1 n 1 n e n+k = e (α n+k+1 γ) n k + Thus (48) f = 1 + l=1 i 1,...,i l =l j=1 e n+k u n+k, k=0 where e n+k defined in (47). By (16) and (46) we obtain fu n = f j,n u j = 0 and hence j=n (49) fu i = 0 for all i n. ) ) l (α k+1+ij γ) e k+l. We show that f given by (48) is idempotent. By (22), for every k 0, we can write f n+k,m = m n f n+k r,n P n+k r,n, for n m n + k. Hence by (46) r=0 (50) f n+k,m = 0 for all n m n + k and k 0. j Using (45) and (50) it follows easily that e i f j,i = e j for all j 0 and hence f is an idempotent. Thus we can write (51) K S [[X]] = K S [[X]](f) K S [[X]](1 f), where by (49) K S [[X]](f) = {( n 1 identity. Define φ : K S [[X]](f) K[X]/ < X γ > n by Since n 1 n 1 φ(( a i u i a i u i )f a i K} is a ring with f as n 1 a i u i )f) = a i u i K[X]/ < X γ > n. = 0 if and only if all a i are equal to zero, by (45) it follows that φ is well defined and one to one. Because by Lemma 1 n 1 γ i X i K[X] can be uniquely written as n 1 a i u i we obtain that φ(( n 1 a i u i )f) = n 1 a i u i = n 1 γ i X i and φ is bijective. It is obvious

THE STRUCTURE OF A RING OF FORMAL SERIES 13 that φ is a K-linear mapping. Because for i n u i = (X γ) i, by (49) we have n 1 φ(( n 1 a i u i )f)(( n 1 n 1 c i u i = c i u i + n 1 b i u i )f) = φ(( 2n 2 n 1 d i u i = ( 2n 2 c i u i + n 1 a i u i )( n 1 n 1 = φ(( a i u i )f)φ(( a i u i )f), n 1 d i u i )f) = φ(( n 1 b i u i ) = ( where c k = k a i b k i and d i K. Hence we obtain that (52) K S [[X]](f) = K[X]/ < X γ > n = K[X]/ < X n >. c i u i )f) = n 1 a i u i )( Also by (49) K S [[X]](1 f) = { a i u i a i K} is a ring with 1 f as identity element. Define ψ : K S [[X]] K S [[X]] by ψ( a i u i ) = g( a n+i v i ) K S [[X]], where g = (X γ) n = u n and v i = i j=1 b i u i ) (X β j ). Since in K S [[X]] g i,i 0, by Theorem 1 g is unit in K S [[X]] and ψ is well defined and bijective. Let h = a i u i, h = b i u i be two elements of K S [[X]](1 f). Then obviously ψ( a i u i + b i u i ) = ψ( a i u i )+ψ( b i u i ) and ψ(u n a i u i ) = ψ( c i u i ) = g( c n+i v i ) = g(g a n+i v i ) = ψ(u n )ψ( a i u i ), where c i K. Hence by induction (53) ψ(u k a i u i ) = ψ(u k )ψ( a i u i ), for every k n. Because ψ is continuous with respect to the corresponding norms defined in (12), by (53) we have ψ(h h ) = ψ( a i h u i ) = ψ(a i u i )ψ(h ) = ψ( a i u i )ψ( b i u i ) = ψ(h )ψ(h ). Hence (54) K S [[X]](1 f) = K S [[X]]

14 GHIOCEL GROZA, AZEEM HAIDER AND S. M. ALI KHAN and by (51),(52) and (54) we have (55) K S [[X]] = K[X]/ < X n > K S [[X]]. Theorem 4. Suppose S = {α n } n 1 is a sequence of elements of K such that a ) I S is a finite set; b ) S has canonical form. Then K S [[X]] = s K[X]/ < X n i > K S [[X]], where S is a purely periodic sequence. Proof. Since α 1 = α 2 =... = α n1 α i, for all i > n 1, then by Lemma 6 we have K S [[X]] = K[X]/ < X n 1 > K S (1)[[X]] where S (1) = {α n1 +i} i 1. Since in S (1) α n1 +1 =... = α n1 +n 2 α i, for all i > n 1 + n 2, then again by Lemma 6 we have K S [[X]] = K[X]/ < X n 1 > K[X]/ < X n 2 > K S (2)[[X]], where S (2) = {α n1 +n 2 +i} i 1. By recurrence we obtain the theorem. Corollary 1. If I S is a finite set, then K S [[X]] is a principal ideal ring. Proof. If S is a purely periodic sequence by [2] Theorem 2.3 it follows that K S [[X]] is a noetherian ring. By Theorem 4 and Remark 1 this is true for every S with I S a finite set. Since by Theorems 2 and 3 all its maximal ideals are principal, K S [[X]] is a principal ring (see for example Lemma 2 from [5]). Corollary 2. If I S is a finite set, for each i I S, < X α i > j is a prime ideal in K S [[X]]. Proof. By Corollary 1 K S [[X]] is a principal ring. Then X α i > j =< h (i) >, h (i) K S [[X]] for all i I S. Suppose there exists f, g K S [[X]] such that f < h (i) > and g < h (i) > but fg < h (i) >. Then there exist two non-negative integers n 1 and n 2 such that (56) f = (X α i ) n 1 f (57) g = (X α i ) n 2 g, where f, g K S [[X]] but f, g < X α i >. Hence by Lemma 5 (58) f g < X α i >. <

THE STRUCTURE OF A RING OF FORMAL SERIES 15 Now by (56)-(58) we obtain fg = (X α i ) n 1+n 2 f g where fg belong to n 1 +n 2 < X α i > j but fg n 1+n 2 +1 < X α i > j. Hence fg < h (i) >, a contradiction which implies that < h (i) > is the prime ideal in K S [[X]] for all i I S. In the next lemma we give the structure of K S [[X]] when S is a purely periodic sequence. Lemma 7. If S = {α n } n 1 is a purely periodic sequence of elements of K, having the smallest period m N, then K S [[X]] = m K[[X]]. Proof. Suppose (59) M = m M i =< u m >, where by Lemma 5 for each i I S, M i =< X α i > is a maximal ideal in K S [[X]]. Since m is the smallest period of a purely periodic sequence we have u k m = u km for all k N and (59) implies that M j =< 0 >. We consider on K S [[X]] the M-adic topology (see for example [9], Ch- VIII) defined by means of (60) ω(f g) = sup{k f g M k }, where f, g K S [[X]] and the distance (61) d(f, g) = e ω(f g), e R, e > 1, which satisfies the ultrametric inequality. We show that K S [[X]] is complete with respect to M-adic topology which in this case coincides with the topology defined by the norm given in (12). Consider {f n } n 1, where f n = a (n) i u i, a Cauchy sequence of elements of K S [[X]]. Then for each ɛ > 0 there exists n 0 (ɛ) N such that d(f n, f n+1 ) < ɛ for all n n 0 (ɛ). Thus for each q N there exists n (q) N such that ω(f n f n+1 ) > q for all n n (q) implies a (n) i = a (n (q)) i for each i q and n n (q). We take f = a i u i K S [[X]], where a q = for each q 0. Then, for all q 0, ω(f n f) > q for each n max{n (0), n (1),..., n (q)}. Hence f K S [[X]] is the limit of the sequence {f n } n 1, which implies that K S [[X]] is complete. By Theorem a (n (q)) q

16 GHIOCEL GROZA, AZEEM HAIDER AND S. M. ALI KHAN 3 and Corollary 2 from [9], Ch-VIII, 8, p.283 we can write (62) K S [[X]] m = R i, where each R i = Let R i = n=0( j i n=0( j i (X α j )) n K S [[X]] is a complete local ring. (X α j )) n =< f i > with f i a nonzero idempotent of K S [[X]]. Define the linear application L fi : K S [[X]] R i by L fi (f) = ff i. Then < h (i) > Ker(L fi ), where < h (i) >= (X α i ) j and f i < h (i) > because f i < h (i) > implies f i M j and hence f i = 0. Since f i < h (i) > there exists p N such that f i = (X α i ) p f i with f i M i. We prove that (63) Ker(L fi ) =< h (i) >. Suppose the contrary that there exists g Ker(L fi ) and g < h (i) >. Then there exists n N such that g = (X α i ) n g with g M i. Because M i is a prime ideal g f i M i and hence (X α i ) p+n g f i 0. This implies that L fi (g) = f i g = (X α i ) p+n g f i 0, a contradiction which implies (63). Hence we obtain (64) R i = KS [[X]]/ < h (i) >. Clearly K R i for each i = 1, 2,..., m and hence each R i is an equicharacteristic ring (see [9], Ch-VIII, 12, p.304). Now by (64) and Corollary 2 each < h (i) > is the prime ideal and R i is an integral domain. By Corollary 1 and (62) each R i is principal local ring. Then each ideal in R i is a power of Mi, where M i is the unique maximal ideal in R i. Since the dimension of R i is one and M i is generated by one element each R i is regular (see [9], Ch-VIII, 11, p.301). Now using Corollary from [9], Ch-VIII, 12, p.307, we obtained that each R i is isomorphic to K i [[Y i ]], where in our case K i = R i / M i = KS [[X]]/M i. Since K S [[X]] is the completion of ring of polynomials K[X] with respect to filtration defined by the ideals < u n >, n N, K S [[X]]/M i = K[X]/ < X αi > = K and we obtain (65) R i = K[[X]]. Now by (65) and (62) it follows that K S [[X]] = m K[[X]].

THE STRUCTURE OF A RING OF FORMAL SERIES 17 Theorem 5. Suppose S = {α n } n 1 is a sequence of elements of K such that I S is a finite set. If s, m, n 1,..., n s are the invariants of S, then K S [[X]] = s K[X]/ < X n i > m K[[X]]. Proof. By Remark 1 we may suppose that S has canonical form. Now the theorem follows by Theorem 4 and Lemma 7.. Acknowledgements The paper was done at the ASSMS GCU Lahore, Pakistan, and was supported by a grant of Higher Education Commission of Pakistan. References [1] Y. Amice, Interpolation p-adique, Bull. Soc. Math. France, Paris, 92(1964), p.117-180. [2] G. Groza, A. Haider, On the ring of Newton interpolating series over a local field, Math. Rep., Bucur. 9(59), No. 4 (2007), 343-356. [3] G. Groza and N. Pop, Approximate solution of multipoint boundary value problems for linear differential equations by polynomial functions, J. Difference Equ. Appl., 14 (2008), No. 12, 1289-1309. [4] G. Groza and N. Pop, A numerical method for solving of the boundary value problems for ordinary differential equations, Result. Math., 53 (2009), No. 3-4, 295-302. [5] Y. Hinohara, On semilocal OP-rings, Proc. Amer. Math. Soc.32, (1972), 16-20. [6] Y. Martin, Sur les séries d interpolation, Ann. Sci. École Norm. Sup., 66 (1949), sér. 3, 311-366. [7] A. Robert, A course in p-adic Analysis, Springer-Verlag, New York, 2000. [8] A. Shidlowski, Transcendental Numbers, de Gruyter, 1989. [9] O. Zariski and P. Samuel, Commutative Algebra, (vol.2), Springer-Verlag, New York, 1960.