COEFFICIENT SUBRINGS OF CERTAIN LOCAL RINGS WITH PRIME-POWER CHARACTERISTIC

If R is a local ring whose radical J(R) is nilpotent and R/J(R) is a commutative field which is algebraic over GF(p), then R has at least one subring S such that S w*=,S,, where each S, is isomorphic to a Galois ring and S/J(S) is naturally isomorphic to R/J(R). Such subrings ofR are mutually isomorphic, but not necessarily conjugate in R.

s is a divisor of r.Conversely, if s is a divisor of r, then there is a unique subring of GR(p",r) which is isomorphic to GR (p", s ).
The following lemma is easily deduced from 16, Theorem 3 (I)] and its proof.
LEMMA 1.1.Let R be a finite local ring of characteristic p" whose residue field is GF(pr).If a e R* satisfies o (a) p r-1, then the subring (a) of R is isomorphic to GR (p", r).
A ring R will be called an IG-ring if there exists a sequence {R,}7= of subrings of R such that R, R,. ,, R, GR(p",r,) (i > 1) and R '= R,, where {r,}= is a sequence of positive integers such that r, r, l(i > 1).If R is an IG-ring described above, then R, is the only subring of R which is isomorphic to GR(p",r,).So we can write R =wT'=GR(p",r,).
Let p be a prime, n a positive integer and r < r < an infinite sequence of positive integers such that r, r, .By the fact we observed above, there exists a natural embedding j-! I+l t',+'GR(p",r,)---GR(p",r,/)foreachi> 1. Letusputt,=idaR(p,,.r,landt,=r:_ t_2o...ot, for <i <j.Then we see that {GR(p",r,),tJ,} is an inductive system.The ring R =lim_GR(p",r,) is an IG-ring.Conversely, any IG-ring can be constructed in this way.An IG-ring R 7'= GR (p", r,) is a Galois ring if lRI is finite.When A is a ring with 1, a subring SofA is called an IG-subring of A if S is an IG-ring.
(II) If e is a positive integer such that < e < n, then R/p'R is naturally isomorphic to the IG-ring w*= GR (p e, r,).
(III) byp. (IV) R is a proper homomorphic image of a discrete valuation ring whose radical is generated Any ideal of R is of the form peR(O < e < n).R is self-injective.
(III) Let us put K 7'= GF(p ').Let W,(K) be the ring of Witt vectors over K of length n (see [15,  Chapter II,6] or [5,Kapitel II,10.4].By (I) and [5, Kapitel II, 10.4], both R and W,(K) are elementary complete local rings (in [14], elementare vollstandige lokale Ringe)of charac- teristic p" whose residue fields are K.Since an elementary complete local ring is uniquely determined by its characteristic and residue field (see [14, Anhang 2]), we see that R is isomorphic to W,(K).Let W(K) be the ring of Witt vectors over K of infinite length.By [7, Chapter V, 7], W(K) is a discrete valuation ring whose radical s generated by p. Since W(K) s the projective limit of {W,,(K)}'= (see [15,Chapter II,6]), W,,(K) is a homomorphic image of W(K).
(V) Clear from (III), since any proper homomorphic image of a principal ideal domain is self-injectlve.
Let {rl}*__ be an infinite sequence of positive integers such that r and rlr/(l > 1), and S 7'-_ GR(p", r) be an IG-ring of characteristic p".Let n n > n2 > > n, be a nonincreasing sequence of positive integers.Let us put Sj wT--t GR(p' 'j, r) for < j < t.Let " S ---> S be the natural homomorphism followed by the isomorphism S/p"JS--S of Proposition 1.2 (II).Let us put U(S;n,,n2,.. nt) {(,j)E(S) Ic,jEp' ifi>j} It is easy to see that U(S;n,n2,.. n,)forms a subring of (S)t,.Let M(S;n,n rtt) denote the set of all t matrices (a,), where a, Sj, and a, p'-"S for > j.Let @ be the mapping of U(S;n,nz n,) onto M(S;n,n n,) defined by (,) t-(a,) where a,j (c,).It is easy to check that addition and multiplication in M(S;nt,n2 n,) can be defined by stipulating that @ preserves addition and multiplication.Following [17], we call M(S ;n,nz n) a ring of Szele matrices over S.
LEMMA 1.3.(cf. 17,Lemma 2.1 ]) Let R be a ring with which contains an IG-subring S of characteristic p".If R is finitely generated as a left S-module, then there exists a nonincreasing sequence n n > nz > > nt of positive integers such that R is isomorphic to a subring of M(S;nl,n nt).PROOF.By Proposition 1.2 (V), there exists a submodule N of R such that R S @ N as left S-modules.By Proposition 1.2 (III), there are a discrete valuation ring W and a homomorphism of W onto S. By defining ay=(a)y (a W, y6N), N is a finitely generated W-module.Since a finitely generated module over a principal ideal domain is a direct sum of cyclic modules, there exist Yt, Y y E N such that N @= Wy,.Let s + 1, x and x, y,_ (2 < < t).Then we get R @= Sx,.Let Sx, S/p"'S as S-modules (n n).Without loss of generality, we may assume nl > n2 > > n,.For each a R, we can write x,a Y ,x (, _ S) Since O=p x,a ,7: p ct, x -rttS by Proposition 1.2 (IV), x, p' if > j.As ,i s uniquely determined modulo p'S by a, we can define :R ---) M(S;n,n n) by a I-) ((o,)).It is easy to see that 1/is an injective ring homo- morphism.

2.
Let G be a group, and N a normal subgroup of G. Let p :G --) H G/N be the natural homo- morphism.A monomorphism k H --G will be called a right inverse of 9 if 9 k idn.If k is a right inverse of p, then G is a semidirect product of N and .(H).
The following lemma is a variation of Schur-Zassenhaus theorem 13, Chapter 9, 9.1.2].LEMMA 2.1.Let G be a group, and N a normal subgroup of G. Let p" G -4 H G/N be the natural homomorphism.Assume that N s locally finite, and there exists a sequence {H,}7'= of finite subgroups of Hsuch that H, cH,+(i > I).v.)7":H,=H and.for any a e N and any > l,o(a) and [H,[ are copnme.Then: (I) There exists a right nverse 'H -G of p.
(III) There exists a unique right inverse of 9 if and only f G is a ndpotent group.
PROOF.(I) For each x e H, we can choose an element g, of G such that 9(g)=x.As G is locally fimte (see [13, Chapter 14, 14.3.1]), the subgroup G of G generated by {g,}., is finite, and p 1, is a homomorphism of G onto H,.Let us put N =Ker(9 ],).Since [N[ and ]H[ are coprime, by Schur-Zassenhaus theorem 13, Chapter 9, 9.1.2],there exists a right inverse .Ht ----) Gi of 9 Ic,, Next, let {g,'}, .be a set of elements of G such that 9(g,') y for any y H, and {g} n, {g,'},, ," Let G2 be the finite subgroup of G generated by {g,'}, , . . .Then 9 1: is a homomorphism of G2 onto H2.By [13,Chapter 9,9.1.3],there exists a complement subgroup L of N=Ker(p I) in G2 such that L ,(H).The mapping Lz'Hz---)Gz defined by H G/N bN_ b(b L) is a right inverse of 9 12-For any a H L(a)-.l(a) NoL { }, hence we see L In,= k.Continuing this process inductively, we get a sequence G G2 c of finite subgroups of G and a sequence {k,}'__ of right inverses k," H, -G, of 9 1, such that .I n, . ,for any _< < j.Then .lim_ %, H wT__ H, --G is a right inverse of 9.
(II) can also be proved in the same way by starting from It" H,,, --It'(H,).
(III) Assume that ."H --G is the unique right inverse of p. Then G is a semidirect product of N and ,(H).We shall show that this is the direct product.Suppose that there exist c e N and z e H such that c.(z) .(z)c.Let us define It'H --) G by It(b) z-k(b)z.Then It is a right inverse of p different from ,, which contradicts our hypothesis.So G is the direct product ofNand .(H).Hence G is nilpotent.
Conversely, let us suppose that G is nilpotent, and .and Ix are right inverses of 19.For each > 1, let G, be the subgroup of G generated by k(H,)w It(H,).Then 9 1, is a homomorphism of G, onto H,.
Both ,(H,) and It(H,) are complement subgroups for N, Ker(p I,) in G,.Since G, is a finite nilpotent group, for each prime divisor q of G,I, G, contains a unique q-Sylow subgroup.Each G, is the direct product of such Sylow subgroups.As n, and N, are coprime, we have X(H,) It(H,).So , In,= It In,.
Since this holds for each > 1, we see .It.
(IV) Let L be the finite subgroup of G generated by It'(H) t.) It"(H,).Then 9 It. is a homomorphism of L onto Hm.Since lKer(9 Iz)l IN LI and n,,I are coprime, by Schur-Zassenhaus theorem, It' (H,) and It"(H,,,) are conjugate in L.
Let G, N, H and p" G H be as in Lemma 2.1.We say that G has property (GC) with respect to N if, for any two right inverses It and v of 9, It(H) and v(H) are conjugate in G.If H is finite, then by Lemma 2.1 (IV), G has the property (GC) with respect to N.
Let R be a ring wth 1.Let S be a subring of R, and I J(R)S.The homomorphism of S/I to R/J(R) defined by a + I a +J(R)(a e S) is in.lective.We shall say that S/I is naturally isomorphic to R/J(R) if this homomorphism is onto.If S is a local subring of a local ring R and if J(S) is nilpotent, then J(S)= J(R)S.Now we shall state main theorems of this section, which generalize the result of R. Raghavendran [9, p. 373, Theorem XIX.4].THEOREM 2.2.Let R be a local ring with radical M. Assume that M is nilpotent, and K R/M is a commutative field of characteristic p (p a prime) which is algebraic over GF(p).Then there exists an IG-subring S of R such that S/pS is naturally isomorphic to K.
PROOF.Since K is algebraic over GF(p), KI is either finite or countably infinite.So there exists a sequence {K,}*= of finite subfields of K such that K, c K, ,(i > 1) and 7_-, K, K. Let K, GR(p r,).
The natural homomorphism rt R --K induces a group homomorphism x* : [R. of R* onto K*.Each (1 +M')/(I +M'*) is isomorphic to the additive group M'/M'+l.As pM' M'/, the order of each element of + M Kern* is a power ofp.Furthermore, K* 7'= K,*, where ]K,*[ p is coprime to p. So, by Lemma 2.1 (I), there exists a right inverse : K* --) R* of rt*.For each > 1, let z, be a generator of K,*.By Lemma 1.1, the subring S, (X(ot,)) of R is isomorphic to GR(p",r,), where p" is the characteristic of R. Consequently, S (k(K*)) 7=S, is an IG-subring of R, and SIpS is naturally isomorphic to K.Such a subring S of R stated in Theorem 2.2 will be called a coefficient subring of R. When R is a commutative local ring satisfying the assumption of Theorem 2.2, S coincides with the subring described in 11, p. 106, Theorem 31.1 ].
Let R, M, S and K t3*= GF(p ') be as in Theorem 2.2, where {r,}7= is a sequence of positive integers such that r, r, t(i > 1).Let p" be the characteristic of R. Let S" be another coefficient subring of R. From what was stated in 1, S' 7'--GR (p", r,), which is isomorphic to S. By Proposition 1.2 (V), there exists a left S'-submodule N of R such that R S" N as left S'-modules.
Ifk K* --> R * is a right inverse of x*, then by the proof ofTheorem 2.3, S Q,,(K*)) is a coefficient subring of R.
Summarizing the above, we obtain the following theorem.THEOREM 2.3.Let R be a local ring with radical M. Assume that M is nilpotent, and K R/M is a commutative field of characteristic p (p a prime) which is algebraic over GR (p).Let n* R* ---) K* be the group homomorphism induced by the natural ring homomorphism :" R K. Then" (I) If S' is a coefficient subrlng of R, then there exists a S'-submodule N of R such that R S' @ N as left S'-modules.
(II) All coefficient subnngs of R are isomorphic.
(III) If 2,." K* --R* is a right inverse of n:*, then S ()(K*)) is a coefficient subring of R. Con- versely, if S is a coefficient subnng of R, then there exists uniquely a right inverse such that S ((K*)).
(IV) All coefficient subrings of R are conjugate in R if and only if R* has property (GC) with respect to + M.
With the same notation as in Theorem 2.2, M/M is regarded as a left K-space by the operation a x ax (-d K R/M,-e M/M2).THEOREM 2.4.Let R be a local ring with radical M. Assume that M is nilpotent, and K R/M is a commutative field of characteristic p (p a prime) which is algebraic over GF(p).Let Sbe a coefficient subring of R. Then R is finitely generated as a left S-module if and only if M/M is a finite dimensional left K-space.In this case, there exists a finitely generated left S-submodule N of M such that R S N as left S-modules, and there exists a nonincreasing sequence nl > ii _> _> n, of positive integers (p"' is the characteristic of R) such that R is isomorphic to a subring of M (S ;n,n2 n,).
PROOF.Assume that R is finitely generated as left S-module.Then R is a Noetherian left S-module, since S is a Noetherian ring by Proposition 1.2 (IV).As M is a left S-submodule of R, M is a finitely generated left S-module.This implies that M/M is a finite dimensional left K-space.
Conversely, let us assume that M/M is a finite dimensional left K-space.Let t.o be the nilpotency index of M. Let x,x:, xa be elements of M whose images modulo M form a K-basis of M/M2.As S/pS is naturally isomorphic to K, any element of y of M is written as y=a,=a,x,+ y" (a, S,y' M2).
Each x,b is written as x,b Z= c, xa + w,j" (q,j e S, w,' e M2).
So we see that any element v' of M can be written as v'='[4=a,x,x+v'" (a, S,v" M3).
Continuing in this way, we see that any element of M is written as S-coefficient line combination of distinct products ofor fewer x,'s.So M is a finitely generated left S-module.Also K R/M is a finitely generated left S-module, hence R is a finitely generated left S-module.Now suppose that R is finitely generated as left S-module.By Theorem 2.3 (I), there exists a finitely generated left S-submodule N' of R such that R S N' as left S-modules.By Proposition 1.2 (III), there exist a discrete valuation ring V and a homomohism of V onto S. Defining ay (a )y (a V, y N'), we can regard N' as a left V-module.Then there exist x,x2 xt N" such that N'= @i= Vx, )I= ,Sx,.By putting x0 1, we get R @_-oSx,.Let c,,c2 c, be elements of S such that c, x, under the natural homomorphism :" R --K.Let us put Y0 and y, x,-c, for <i <t.Then y, e M(1 <i < t)andR =l=oSx,=i=oSy SoN i__ Sy, has the desired property.The last statement is immediate from Lemma 1.3.
Let R be a local ring described in Theorem 2.2.Then R may have more than one coefficient subring.Concerning this subject, first we can state the following.THEOREM 3.1.Let Tbe an IG-ring of characteristic p" different from GR(p", 1).Then, for any infinite cardinal number %, there exists a local ring R such that (1) M J(R) is nilpotent, (2) K R/M is a commutative field of characteristic p (p a prime) which is algebraic over GF(p), (3) coefficient subrings of R are isomorphic to T, (4) all coefficient subrings of R are conjugate in R, and (5) % is the number of all coefficient subrings of R.
PROOF.Let T=w7=GR(p",r,), where {r,}*= is a sequence of positive integers such that r, r, (i > 1).Let K T/pT and " T --K be the natural homomorphism.As K is a proper extension of GF(p), there exists an automorphism of K different from idK.Let be the automorphism of T which induces t modulo pT (see Proposition 1.2 (VI)).LetA be a set of cardinality %, and V , ,T be a free T-module.The abelian group T V together with the multiplication (a,x) (a',x') (aa',ax" + (a')x) forms a ring, which we denote by R. Let n:'R K be the homomorphism defined by (a,x) t-'(a), and M Kerry.As R/M K and M l= O, R is a local ring with radical M whose residue field is K.By Theorem 2.3 (1117, there exists a one-to-one correspondence between the set of all coefficient subrings of R and the set Yof all right inverses of :* = IRo" R* --K*. By the embedding T 9 a -(a, 0) R, T is regarded as a coefficient subring of R. So, by Theorem 2.3 (liD, there exists a right inverse ."K* R* of :* such that (,(K*)) T. Since K wT--t GF(pr'), there exists a number j > such that is not the identity on GF(p ).Let 7be a generator of GF(p ')*, and c ,(?).It is easy to see that, for any z V, R* h (c, z) is of multiplicative order p 1. So, for each z V, we can define a group homomorphism tz" GF(pr)* R* by -(c,z)'.By Lemma 2.1 (II), we can extend t.tz" to l.t Y.If V z,z2 and Z :g: Z2, then l.tt : ll, z2.So YI > VI %.
Let K w7=GF(p '), where {r,}7: is a strictly ncreasing sequence of positive integers such that r, r, (i > 1).Let {or,}'= be automorphisms of K such that or, is not the identity on GF(p") (i > 1) and, forj < i, or, is the identity on GF(pr).Let V 0)7= tKx be a left K-vector space with basis {x,}7__ i.We can regard V as a (K,K)-bimodule by defining (E, c,x,)a , , c,or,(a)x, (Y, c,x, e V,a K).
The abelian group R K 9 V together with the multiplication (a, y) (b, z) (ab, az + yb) (a, b K, y, z V) forms a local ring with radical M (0, V), which satisfies the assumption of Theorem 2.3.The homo- morphism rt:R K defined by (a,x)a gives the isomorphism R/M--K.The subring S { (a, 0) a e K} of R is a coefficient subring of R.
where each K,(1 < < d) is a commutative field of characteristic p (p a prime) which is algebraic over GF(p).Then there exists a subring T of R which satisfies the following.
(i) R T N (as abelian groups), where N s an additive subgroup of R. (ii) T is isomorphic to a finite direct sum of matrix rings over IG-rings. (iii) J(T)=TJ(R)=pT.
(iv) T/pT is naturally isomorphic to R/J(R).
As e,Te, is an IG-ring which is naturally isomorphic to e,Re,/e, J(R)e,, so e,Te, is a coefficient subring of e,Re,.
Similarly, fT'f is a coefficient subring of fRf.

Call for Papers
Space dynamics is a very general title that can accommodate a long list of activities.This kind of research started with the study of the motion of the stars and the planets back to the origin of astronomy, and nowadays it has a large list of topics.It is possible to make a division in two main categories: astronomy and astrodynamics.By astronomy, we can relate topics that deal with the motion of the planets, natural satellites, comets, and so forth.Many important topics of research nowadays are related to those subjects.By astrodynamics, we mean topics related to spaceflight dynamics.
It means topics where a satellite, a rocket, or any kind of man-made object is travelling in space governed by the gravitational forces of celestial bodies and/or forces generated by propulsion systems that are available in those objects.Many topics are related to orbit determination, propagation, and orbital maneuvers related to those spacecrafts.Several other topics that are related to this subject are numerical methods, nonlinear dynamics, chaos, and control.
The main objective of this Special Issue is to publish topics that are under study in one of those lines.The idea is to get the most recent researches and published them in a very short time, so we can give a step in order to help scientists and engineers that work in this field to be aware of actual research.All the published papers have to be peer reviewed, but in a fast and accurate way so that the topics are not outdated by the large speed that the information flows nowadays.
Before submission authors should carefully read over the journal's Author Guidelines, which are located at http://www .hindawi.com/journals/mpe/guidelines.html.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http://mts.hindawi.com/according to the following timetable: