On linear difference equations over rings and modules

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PII. S0161171204302085 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON LINEAR DIFFERENCE EQUATIONS OVER

RINGS AND MODULES

JAWAD Y. ABUHLAIL

Received 8 February 2003

We develop a coalgebraic approach to the study of solutions of linear diﬀerence equations over modules and rings. Some known results about linearly recursive sequences over base ﬁelds are generalized to linearly (bi)recursive (bi)sequences of modules over arbitrary com-mutative ground rings.

2000 Mathematics Subject Classiﬁcation: 16W30, 39A99.

1. Introduction. Although the theory of linear diﬀerence equations over base ﬁelds is well understood, the theory over arbitrary ground rings and modules is still under development. It is becoming more interesting and is gaining increasingly special impor-tance mainly because of recent applications in coding theory and cryptography (e.g., [10,15]).

In a series of papers, Taft et al. (e.g., [17, 22, 24]) developed a coalgebraic aspect for the study of linearly recursive sequences over fields. Moreover, Grünenfelder et al. studied in [8,9] the linearly recursive sequences overfinite-dimensionalvector spaces. Linearly recursive (bi)sequences over arbitrary rings and modules were studied inten-sively by Nechaev et al. (e.g., [16,20,21]); however, the coalgebraic approach in their work was limited to the field case. Generalization to the case of arbitrary commuta-tive ground rings was studied by several authors including Kurakin [12,13, 14] and eventually Abuhlail, Gómez-Torrecillas, and Wisbauer [4].

In this paper, we develop a coalgebraic aspect for the study of solutions of linear diﬀerence equations overarbitrary rings and modules. For some of our results, we assume that the ground ring is Artinian. Besides the new results, this paper extends results in [4] and “Kapitel 4” of my doctoral thesis [2]. A standard reference for the theory of linearly recursive sequences over rings and modules is the comprehensive work of Mikhalev et al. [16]. For the theory of Hopf algebras, the reader may refer to any of the classical references (e.g., [1,19,23]).

By R we denote a commutative ring with 1R≠0R and withU (R)= {r ∈R|r is invertible}the group ofunits ofR. The category ofR-(bi)modules will be denoted by ᏹR. For anR-moduleM, we call anR-submoduleK⊂Mpure(in the sense of Cohn) if for everyR-moduleN, the induced mapιk⊗idN:K⊗RN→M⊗RNis injective.

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A-moduleMtheﬁnite dualrightA-module

M◦:=f∈M∗|Ke(f )⊃IM

for someA-idealIwithA/Iﬁnitely generated overR. (1.1)

ByN(resp.,Z) we denote the set of natural numbers (resp., the ring of integers). More-over, we setN0:= {0,1,2,3, . . .}. For ann×nmatrixMoverR, we denote the

charac-teristic polynomial byχ(M). The identity matrix of ordernoverR is denoted byEn. For anm×nmatrixAand ak×lmatrixB, theKronecker product(tensor product) of AandBis themk×nlmatrix

A⊗B:=       

a11·B a12·B ··· ··· a1n·B a21·B a22·B ··· ··· a2n·B

..

. ... ... ... ... am1·B am2·B ··· ··· amn·B

     

. (1.2)

2. Preliminaries. LetMbe anR-module and

M[x]:=M x1, . . . , xk, M x,x−1:=M x1, x1−1, . . . , xk, xk−1

. (2.1)

We consider thepolynomial ringR[x]and thering of Laurent polynomialsR[x,x−1_]_as

commutativeR-algebras with the usual multiplication and the usual unity. For everyR -moduleM,M[x](resp.,M[x,x−1_]_{) is an}_R[_x_]_{-module (resp., an}_R[_x_,_x−1_]_{-module) with}

action induced from theR-module structure onM and we have, moreover, canonical R-module isomorphisms

M[x]M⊗RR[x]M(N k

0)_, _M _x_,_x−1_M⊗_R_R _x_,_x−1_M(Zk)_. _(2.2)

Forn=(n1, . . . , nk)∈Nk0 (resp.,z=(z1, . . . , zk)∈Zk), we setxn:=xn11, . . . , x

n_k k (resp.,

xz_:₌_xz1

1 , . . . , x

zk k ).

2.1. LetM be anR-module,l=(l1, . . . , lk)∈Nk0, and consider thesystem of linear

diﬀerence equations(SLDE)

xn+(l1,0,...,0)+ l1

i=1

p(1,l1−i)(n)xn+(l1−i,0,...,0)=g1(n),

xn+(0,l2,0,...,0)+ l2

i=1

p(2,l2−i)(n)xn+(0,l2−i,0,...,0)=g2(n),

.. .

xn+(0,...,0,lk)+ lk

i=1

p(k,l_k−i)(n)xn+(0,...,0,lk−i)=gk(n),

(2.3)

where thepjl’s areR-valued functions and thegj’s areM-valued functions deﬁned for alln∈Nk

0. If thegj’s are identically zero, then (2.3) is said to be ahomogenousSLDE.

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2.2. For anR-moduleMandk≥1, let

᏿kM :=

u:Nk

0 →M

MNk0 _(2.4)

be theR-module ofk-sequencesoverM. IfM(resp.,k) is not mentioned, then we mean thatM=R(resp.,k=1). Forf (x)=iaixi∈R[x]andw∈᏿Mk, deﬁne

f (x) w=u∈᏿k

M , u(n):=

i

aiw(n+i)∀n∈Nk0. (2.5)

With this action,᏿Mk is anR[x]-module. For subsetsI⊂R[x]andL⊂᏿ k

M , consider the annihilator submodules

An_᏿k M (I)=

w∈᏿kM |f w=0 for everyf∈I

,

AnR[x](L)=

h∈R[x]|h u=0 for everyu∈L. (2.6)

Note that An_᏿k_M(I)⊂᏿k

M is anR[x]-submodule and AnR[x](L) R[x]is an ideal.

2.3. A polynomialf (x)∈R[x]is called monic if its leading coeﬃcient is 1R. For every monic polynomialf (x)=xl₊_al−

1xl−1+ ··· +a1x+a0∈R[x], thecompanion

matrixoffis deﬁned to be thel×lmatrix

Sf :=          

0R 0R ··· 0R −a0

1R 0R ··· 0R −a1

0R 1R ··· 0R −a2

..

. ... ... ... ... 0R 0R ··· 1R −al−1

         

. (2.7)

In factSf is a matrix that hasf (x)as its characteristic polynomial as well as its mini-mum polynomial (see [11, Theorem 4.18]).

Definition2.1. An idealI R[x]will be calledmonic, if it contains a nonempty subset of monic polynomials

fjxj=xl_jj+a(j)_l j−1x

l_j−1

j +···+a (j)

1 xj+a

(j)

0 |j=1, . . . , k

. (2.8)

In this case, the polynomials (2.8) are calledelementary polynomials and(f1(x1), . . . ,

fk(xk)) R[x] an elementary ideal. A monic polynomial q(x)∈ R[x] is called re-versibleifq(0)∈U (R). An idealI R[x,x−1_]_{will be called}_reversible_{if it contains a}

subset of reversible polynomials{q1(x1), . . . , qk(xk)}.

2.4. LetMbe anR-module. We callu∈᏿kM alinearly recursivek-sequence(resp., a linearly birecursivek-sequence) if AnR[x](u)is a monic ideal (resp., a reversible ideal).

Note that ak-sequenceu∈᏿k

M is linearly recursive if and only if it is a solution of a homogenous SLDE with constant coeﬃcients of the form (2.3). If AnR[x](u)contains a

set of monic polynomials{f1(x1), . . . , fk(xk)}, wherefj(xj)is of ordermj,j=1, . . . , k,

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are calledminimal polynomialsofuandn:=(n1, . . . , nk)is called therank ofu. The

subsetsᏸkM ⊆᏿kM of linearly recursivek-sequences andᏮkM ⊆᏿kM of linearly birecur-sivek-sequences are obviouslyR[x]-submodules.

2.5. Thelexicographical linear order()onNk0is deﬁned as follows: fori=(i1, . . . , ik)

andn=(n1, . . . , nk)∈Nk0, we sayinif the ﬁrst number in the sequence of integers

n1+···+nk−i1+···+ik, n1−i1, . . . , nk−ik (2.9)

that is diﬀerent from zero is positive (see [18, page 170]).

LetMbe anR-module,F:= {f1(x1), . . . , fk(xk)} ⊂R[x]a subset of monic polynomials

with deg(fj(xj))=ljforj=1, . . . , k,l:=(l1, . . . , lk),1:=(1, . . . ,1), andIF:=(f1, . . . , fk)

R[x]. Note that the natural order “≤” onN0 induces on Nk0 a partial order and we

deﬁne thepolyhedronΠF=Π(l):= {i∈Nk0|i≤l−1}. Theinitial polyhedron of values

ofω∈᏿k

M is deﬁned asω(ΠF):= {ω(i)|i∈ΠF}. Forl=l1, . . . , lk, the points of the

polyhedronΠFbuild a chain0=i0i1 ··· il−1and we can writeω(ΠF)as aninitial

vector of values(ω(0), ω(i1), . . . , ω(il−1))∈Ml.

Letω∈An

᏿Mk(f1(x1), . . . , fk(xk)), wherefj(xj)is monic forj=1, . . . , nand write, for everyn=(n1, . . . , nk)∈Nk0,

x_jnj=hjxjfjxj+rjxj, degrjxj< lj. (2.10)

If we set

g(n)(x):=

k

j=1

rjxj=

i∈ΠF

a(in)xi, v:=xn ω=g(n)(x) ω, (2.11)

then

ω(n)=v(0)=

i∈ΠF

a(_in)ω(i) for everyn∈Nk

0. (2.12)

Consequently,ωis completely determined by the initial polyhedron of valuesω(ΠF).

Fort∈ΠF, deﬁne the sequenceetF∈An_᏿k

R (IF)with initial polyhedron of valuese

F t(i)=

δi,tfor alli∈ΠF. The sequenceeFl−1is called theimpulse sequenceof An_᏿k R (IF).

3. Examples. We now give some examples of linearly recursive sequences. For more examples, the reader may refer to [16].

Example3.1(geometric progression). LetM be anR-module,m∈M,r ∈R, and considerw∈᏿M given by

w(n):=rnm for everyn∈N0. (3.1)

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Example3.2(arithmetic progression). LetMbe anR-module,{p, q} ⊂M, and con-siderw∈᏿M given by

w(n):=p+nq for everyn∈N0. (3.2)

Thenw∈ᏸM with initial vector(p, p+q)and elementary characteristic polynomial f (x)=(x−1)2_{. If An}_R(q)₌_{0, then}_{f (x)} _{is a unique minimal polynomial of} _w_{. If}

r∈AnR(q), thenfr(x)=(x−1)2₊_{r (x}₋₁₎_{is another minimal polynomial of}_w_.

Remark3.3. An example of a nonrecursive sequence overZis the sequence of prime positive numbers{2,3,5,7, . . .}.

Example3.4. LetE= {f₁(x), . . . , fk(x)} ⊂R[x]be a subset of monic polynomials. (1) LetM be anR-module,ui ∈An_᏿_M(fi) fori=1, . . . , k, and consider u:=u1

·

+ ···+·uk∈᏿k

M deﬁned byu(n)=u1(n1)+ ··· +uk(nk). Thenu∈An_᏿k

M (g1(x1), . . . , gk(xk)), where fori=1, . . . , k,

gixi=   

fixi, fi1R

=0R, fixixi−1R, otherwise.

(3.3)

(2) LetM1, . . . , Mk beR-modules,ui∈An᏿_Mi(fi)fori=1, . . . , k,M:=M1⊕ ··· ⊕Mk,

and consideru∈᏿kM deﬁned byu(n):=(u1(n1), . . . , uk(nk)). Thenu∈An_᏿k

M (g1(x1), . . . , gk(xk)), where thegi’s are deﬁned as in (3.3).

(3) Let ui ∈ An_᏿_R(fi) for i= 1, . . . , k and consider u ∈ ᏿k

R deﬁned by u(n):= u1(n1), . . . , uk(nk). Thenu∈An_᏿k

R (f1(x1), . . . , fk(xk))and

An_᏿k R

f1

x1

, . . . , fkxkAn᏿R

f1

⊗R···⊗RAn᏿R

fk. (3.4)

(4) LetM1, . . . , MkbeR-modules,ui∈An᏿_Mi(fi)fori=1, . . . , k,M:=M1⊗R···⊗RMk,

and consideru∈᏿k

M deﬁned byu(n):=u1(n1)⊗···⊗uk(nk). Thenu∈An_᏿k

M (f1(x1), . . . , fk(xk))and

An_᏿k M

f1

x1

, . . . , fkxkAn_᏿_M 1

f1

⊗R···⊗RAn᏿_Mkfk. (3.5)

4. AdmissibleR-bialgebras and HopfR-algebras. For everyR-coalgebra(C,∆C, εC), there is adualR-algebraC∗:=HomR(C, R)with the so-calledconvolution product mul-tiplication

(f∗g)(c):=fc1

gc2

∀f , g∈C∗, c∈C, (4.1)

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Definition 4.1. Let A be an R-algebra (resp., an R-bialgebra, a Hopf R-algebra). ThenAis called

(1) anα-algebra(resp., anα-bialgebra, aHopf α-algebra) if᏷Ais a ﬁlter andA◦⊂RA is pure;

(2) cofinitaryif᏷Ais afilter and, for everyI∈᏷A, there exists anA-ideal ¯I⊆Iwith A/¯Ifinitely generated and projective.

4.1. LetHbe anR-bialgebra and consider the class ofR-cofiniteH-ideals᏷H. We call HanadmissibleR-bialgebraifHis cofinitary and᏷Hsatisfies the following axioms:

(A1) for allI, J∈᏷H, there existsL∈᏷H, such that∆H(L)⊆Im(I⊗RH)+Im(H⊗RJ), (A2) there existsI∈᏷H, such that Ke

εH⊃I.

We call a HopfR-algebraH anadmissible Hopf R-algebra ifH is coﬁnitary and᏷H satisﬁes (A1), (A2), and

(A3) for everyI∈᏷H, there existsJ∈᏷H, such thatSH(J)⊆I.

Remark_4.2. _{It follows from the proof of [}₃_{, Proposition 4.2.] that every coﬁnitary} R-algebra (resp.,R-bialgebra, HopfR-algebra) is anα-algebra (resp., anα-bialgebra, a Hopfα-algebra). By [2, Lemma 2.5.6.], every coﬁnitary bialgebra (Hopf algebra) over a Noetherianground ring is admissible.

Proposition_4.3_{(see [}₂_{, Propositions 2.4.13 and 2.5.7])}. ₍₁₎_If_A_{is a coﬁnitary} R-algebra, thenA◦is anR-coalgebra. IfHis an admissibleR-bialgebra (resp., an admissible HopfR-algebra), thenH◦is anR-bialgebra (resp., a HopfR-algebra).

(2)LetRbe Noetherian. IfAis anα-algebra (resp., anα-bialgebra, a Hopfα-algebra), thenA◦is anR-coalgebra (resp., anR-bialgebra, a HopfR-algebra).

Proposition_4.4. _Let_A_{be an}_{α-algebra (resp., an}_α_{-bialgebra, a Hopf}_α-algebra), Ba coﬁnitaryR-algebra (resp.,R-bialgebra, HopfR-algebra), and consider the canonical mapσ:A◦⊗RB◦→(A⊗RB)◦. Then

(1) σ is injective,

(2) ifRis Noetherian, thenσis an isomorphism ofR-coalgebras (resp.,R-bialgebras, HopfR-algebras).

Proof. _{(1) The proof is along the lines of the proof of [}₁₄_{, Proposition 5].} (2) The proof is along the lines of the proof of [3, Theorem 4.10].

The proof of [3, Lemma 4.12] can be generalized to prove the following lemma.

Lemma4.5. For any set of reversible polynomials{q₁(x₁), . . . , qk(xk)} ⊆R[x], there is an isomorphism ofR-algebras

R[x]/q1x1, . . . , qkxkR x,x−1/q1(x1, . . . , qkxk. (4.2)

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(2)An idealI R[x,x−1_]_is_{R-coﬁnite if and only if it is reversible. Consequently, every}

R-coﬁniteR[x,x−1_{]-ideal contains an ideal}_{I R[}_¯ _x_,_x−1_]_{such that}_R[_x_,_x−1_]/_¯_I _{is free of}

ﬁnite rank. In particular,R[x,x−1_]_{is coﬁnitary.}

5. Linearly (bi)recursive sequences. In this section, we study thelinearly(bi) recur-sivek-sequencesoverR-modules, whereRis an arbitrary commutative ground ring.

5.1. Let(G, µG, eG)be a (commutative) monoid. Considering the elements of the basis Gasgroup-like elements, the monoid algebraRGbecomes a (commutative) cocommu-tativeR-bialgebra(RG, µ, η,∆g, εg), where

∆g(x)=x⊗x, εg(x)=1R for everyx∈G. (5.1)

IfGis a group, thenRGis a HopfR-algebra with antipode

Sg:RG →RG, x→x−1 _{for every}_x_∈_G. _(5.2)

5.2. Bialgebra structures onR[x]. Consider the commutative monoidGgenerated by{xj|j=1, . . . , k}. ThenR[x]=RGhas the structure of acommutative cocommutative R-bialgebraR[x;g]=(R[x], µ, η,∆g, εg), where µis the usual multiplication, ηis the usual unity, and for alln≥0, j=1, . . . , k,

∆g:R[x]→R[x]⊗RR[x], xjn→xnj⊗xnj,

εg:R[x] →R, x_jn→1R. (5.3)

On the other hand,R[x;p]=(R[x], µ, η,∆p, εp)is acommutative cocommutativeHopf R-algebra, whereµ is the usual multiplication,ηis the usual unity, and for alln≥0, j=1, . . . , k,

∆p:R[x]→R[x]⊗RR[x], xjn→ n

t=0

n t

xjt⊗xjn−t,

εp:R[x] →R, xn_j →δn,0,

Sp:R[x] →R[x], xn

j →(−1)nxnj.

(5.4)

Remarks5.1. (1) LetRbe an integral domain, then it follows by [7, Theorem 1.3.6] that for every setG, the class of group-like elements of theR-coalgebraRGisGitself. Then one can show, as in the ﬁeld case [6], thatR[x;g]andR[x;p]are the only possible R-bialgebra structures onR[x]with the usual multiplication and the usual unity.

(2) TheR-bialgebraR[x;g]has no antipode because the group-like elements in a Hopf R-algebra should be invertible.

The proof of the following result depends mainly on the arguments of [14, Theorem 2].

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Proof. Denote by(R[x],∆, ε)either of the cofinitaryR-bialgebrasR[x;g]andR[x;p]. LetI, J R[x]beR-cofinite ideals and assume without loss of generality thatR[x]/I andR[x]/Jare free of finite rank (seeLemma 4.6). Letβbe a basis of the freeR-module B:=R[x]/I⊗RR[x]/Jand consider theR-algebra morphism ¯∆:=(πI⊗πJ)◦∆:R[x]→ R[x]/I⊗RR[x]/J. Forj=1, . . . , k, letMjbe the matrix of theR-linear map

Tj:B →B, b→∆¯xjb, (5.5)

with respect toβ, andχj(λ)its characteristic polynomial. Thenχj(_∆¯(xj))=0 forj= 1, . . . , k. Since ¯∆is an R-algebra morphism, it follows thatχj(xj)∈Ke(∆¯)=∆−1_(I_⊗

R R[x]+R[x]⊗RJ)forj=1, . . . , k. If we setL:=(χ1(x1), . . . , χk(xk)) R[x], then∆(L)⊆

I⊗RR[x]+R[x]⊗RJ, that is, ᏷R[x] satisﬁes axiom (A1). Note that R[x]/Ke(ε)R, hence᏷R[x]satisﬁes axiom (A2). Consequently,R[x;g]andR[x;p]are admissibleR -bialgebras. Consider now the Hopf R-algebra R[x;p]with the bijectiveantipode Sp. For every ideal I R[x], S−1

p (I) R[x;p] is an ideal and we have an isomorphism ofR-modulesR[x]/S−1

p (I)R[x]/I, hence᏷R[x;p] satisﬁes axiom (A3). Consequently, R[x;p]is an admissible HopfR-algebra. The last statement follows now byProposition 4.3.

If M is an arbitrary R-module, then we have obviously an isomorphism ofR[x] -modules

ΦM:M[x]∗ →᏿kM∗, → n→ m→

mxn, (5.6)

with inverseu[mxn_u(_n_)(m)]_.

Proposition5.3. LetMbe anR-module. Then (5.6) induces an isomorphism ofR[x ]-modules

M[x]◦ᏸkM∗. (5.7)

Proof. _{Consider the}_R[_x_]_{-module isomorphism}_M[_x_]∗ΦM ᏿k_M_∗_{, see (}_5.6_{). Let}∈ M[x]◦. Then there exists anR-coﬁniteR[x]-idealIsuch thatI =0. SoI ΦM()=

ΦM(I )=0, that is,I⊂AnR[x](ΦM()). ByLemma 4.6(1),Iis monic, that is,ΦM()∈ ᏸMk∗.

On the other hand, letu∈ᏸk

M∗. By deﬁnition,J:=AnR[x](u)is a monic ideal and it

follows byLemma 4.6(1) thatJ R[x]isR-coﬁnite. For:=Φ−1

M (u), we haveJ = J Φ−1

M (u)=ΦM−1(J u)=0, that is,∈M[x]◦.

5.3. The coalgebra structure onᏸk_. _By_{Lemma 4.6}_(1),_(R[_x_{], µ, η)}_{is a coﬁnitary}_R -algebra, whereµis the usual multiplication andηis the usual unity. Hence,(R[x]◦, µ◦, η◦)is (byProposition 4.3) anR-coalgebra, where

µ◦:R[x]◦ →R[x]◦⊗RR[x]◦, f→ xis⊗x t j→f

x_isxt_j, s, t≥0, i, j=1, . . . , k,

η◦:R[x]◦ →R, f→f1R

.

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Soᏸk_R[_x_]◦_{has the structure of an}_R_{-coalgebra with counity}

ε_ᏸk:ᏸk →R, u→u(0), (5.9)

and comultiplication described as follows (see [16, Proposition 14.16]).

Letu∈ᏸk_,_{_f₁_(x₁_{), . . . , fk(xk)}_{} ⊆}_An_R[_x_](u)_{a subset of elementary characteristic} polynomials with deg(fj(xj))=lj, andl:=(l1, . . . , lk). So we have for alln,i∈Nk0,

u(n+i)=xi_u₍_n₎₌





t≤l−1

xi_u₍_t₎_·_eF t



(n)=

t≤l−1

xt_u₍_i₎_·_eF

t(n). (5.10)

The comultiplication ofᏸkis given then by

∆ᏸk:ᏸk →ᏸk⊗Rᏸk, u→

t≤l−1

xt_u_⊗_eF

t. (5.11)

Example_5.4. _{Consider the} _{Fibonacci sequence}=₍₀_,₁_,₁_,₂_,₃_,₅_{, . . .)}_{. Clearly,} _is given by

(0)=0, (1)=1, (n+2)=(n+1)+(n) ∀n≥0, (5.12)

that is,∈ᏸZwith initial vector(0,1)and elementary characteristic polynomialf (x)=

x2₋_x₋₁_∈_Z_[x]_{. By (}_5.11_{), one can easily calculate}

∆ᏸZ()=⊗Z(x )+(x )⊗Z−⊗Z. (5.13)

5.4. TheR-bialgebra(ᏸkR ;g). Consider theR-bialgebraR[x;g]. Then᏿kRN k 0

R[x;g]∗is anR-algebra with multiplication given by theHadamard product

∗g:᏿k⊗R᏿k →᏿k, u⊗v→[n→u(n)v(n)], (5.14)

and the unity

ηg:R →᏿k_, ₁

R→ n→1R

for everyn∈Nk

0. (5.15)

By Propositions5.2and5.3,(ᏸkR ;g)R[x;g]◦has the structure of anR-bialgebra with the coalgebra structure described inSection 5.3, the Hadamard product (5.14), and the unity (5.15).

5.5. The HopfR-algebra(ᏸk

R ;p). Consider the HopfR-algebraR[x;p]. Then᏿k RNk0_R[_x_;_p]∗_{is an}_R_{-algebra with multiplication given by the}_{Hurwitz product}

∗p:᏿k⊗R᏿k →᏿k, u⊗v→

n→

t≤n

n

t

u(t)v(n−t)

, (5.16)

and the unity

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By Propositions5.2and5.3,(ᏸk

R ;p)R[x;p]◦ has the structure of a HopfR-algebra with the coalgebra structure described inSection 5.3, the Hurwitz product (5.16), the unity (5.17), and the antipode

S_ᏸk:ᏸk →ᏸk, u→ i→(−1)iu(i). (5.18)

Proposition_5.5_{(see [}₁₄_{, Theorem 3])}. _Let_u_and_v_{be linearly recursive sequences} overRof ordersm,nand with characteristic polynomialsf (x),g(x), respectively. Then (1) u∗gv is a linearly recursive sequence overR of orderm·nand characteristic

polynomialχ(Sf⊗Sg);

(2) u∗pv is a linearly recursive sequence overRof orderm·nand characteristic polynomialχ(Sf⊗En+Em⊗Sg).

Example5.6. LetR be any ring and let{xn}∞_n₌₀,{yn}∞_n₌₀∈᏿_Rbe solutions of the diﬀerence equations

xn₊3−xn+2+xn−1−xn=0; x0=0, x1=1, x2=2;

yn+2−yn+1+yn=0; y0=1, y1=0.

(5.19)

Then{xn}∞_n=0is a linearly recursive sequence over R with characteristic polynomial

f (x)=x3₋_x2₊_x₋_{1 and}_{_yn_}∞

n=0is a linearly recursive sequence overRwith

charac-teristic polynomialg(x)=x2₋_x₊_1.

Notice that

Sf⊗Sg=    

0 0 1

1 0 −1

0 1 1

   ⊗

0 −1

1 1

=           

0 0 0 0 0 −1

0 0 0 0 1 1

0 −1 0 0 0 1

1 1 0 0 −1 −1

0 0 0 −1 0 −1

0 0 1 1 1 1

          

.

(5.20)

Hence, {zn}∞

n=0 := {xn}∞n=0∗g{yn}∞n=0 is by Proposition 5.5 a linearly recursive

se-quence overRwith characteristic polynomial

χSf⊗Sg=x6₋_x5₊_x3₋_x₊₁_, _(5.21)

that is,{zn}∞

n=0is a solution of the diﬀerence equation

[image:10.468.152.316.368.488.2]

zn+6−zn+5+zn+3−zn+1+zn=0 with initial vector(0,0,−2,−1,0,1). (5.22)

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Table5.1

n 0 1 2 3 4 5 6 7 8 9 10

xn 0 1 2 1 0 1 2 1 0 1 2

yn 1 0 −1 −1 0 1 1 0 −1 −1 0

zn 0 0 −2 −1 0 1 2 0 0 −1 0

Example5.7. Consider the sequences{xn}∞_n₌₀and{yn}∞_n₌₀ofExample 5.6. Then

Sf⊗E2+E3⊗Sg=           

0 −1 0 0 1 0

1 1 0 0 0 1

1 0 0 −1 −1 0

0 1 1 1 0 −1

0 0 1 0 1 −1

0 0 0 1 1 2

          

. (5.23)

ByProposition 5.5,{zn}∞

n=0= {xn}∞n=0∗p{yn}∞n=0:= { n

j=0 _n

j

xj·yn−j}∞

n=0is a linearly

recursive sequence overRwith characteristic polynomial

χSf⊗E2+E3⊗Sg=x6−5x5+14x4−25x3+28x2−15x+3. (5.24)

Hence,{zn}∞

n=0is a solution of the diﬀerence equation

zn₊6−5zn+5+14zn+4−25zn+3+28zn+2−15zn+1+3zn=0 (5.25)

with initial vector(0,1,2,−2,−16,−29).

Table 5.2gives the ﬁrst 9 terms of the sequence{zn}∞ n=0.

Table5.2

n 0 1 2 3 4 5 6 7 8

xn 0 1 2 1 0 1 2 1 0

yn 1 0 −1 −1 0 1 1 0 −1

zn 0 1 2 −2 −16 −29 −12 29 0

5.6. Cofree comodules. Let C be an R-coalgebra. A right C-comodule (M, M) is calledcofreeif there exists anR-moduleKsuch that(M, M)(K⊗RC, idK⊗∆C)as rightC-comodules. Note that ifKR(Λ)_{, a free}_R_{-module, then}_M_R(Λ)_⊗

RCC(Λ)as rightC-comodules (this is one reason for the terminologycofree).

As a direct consequence ofLemma 4.6, we get the following corollary.

Corollary5.8. LetMbe anR[x]-module. Then there are isomorphisms ofR[x]◦ -comodules

ᏸkM∗M[x]◦M∗⊗RR[x]◦M∗⊗RᏸkR . (5.26)

In particular,M[x]◦(ᏸk

M∗) is a cofreeR[x]◦-comodule (ᏸ k

[image:11.468.53.417.89.137.2]

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6. Linearly (bi)recursive bisequences. In this section, we consider thelinearly( bi)re-cursivek-bisequencesand thereversiblek-sequencesoverR-modules, whereRis an ar-bitrary commutative ground ring. We generalize the results of [16,17] concerning the bialgebra structure of the linearly recursive sequences over a base ﬁeld to the case of arbitraryArtinianground rings.

6.1. LetM be anR-module,l=(l1, . . . , lk)∈Nk0, and consider the system of linear

bidiﬀerence equations (SLBE):

xz+(l1,0,...,0)+ l1

i=1

p(1,l1−i)(z)xz+(l1−i,0,...,0)=g1(z),

xz+(0,l2,0,...,0)+ l2

i=1

p(2,l2−i)(z)xz+(0,l2−i,0,...,0)=g2(z),

.. .

xz+(0,...,0,lk)+ lk

i=1

p(k,l_k−i)(z)xz+(0,...,0,lk−i)=gk(z),

(6.1)

where thepjl’s areR-valued functions and thegj’s areM-valued functions deﬁned for allz∈Zk_{. If the}_gj_{’s are identically zero, then (}_6.1_{) is said to be a}_homogenous_SLBE. If thepjl’s are constants, then (6.1) is said to be an SLBEwith constant coeﬃcients.

6.2. Bisequences. For anR-moduleMandk≥0, let

᏿k M :=

ν:Zk _→_M_MZk

(6.2)

be the R-module of k-bisequences over M. IfM (resp.,k) is not mentioned, then we meanM=R(resp.,k=1). Forw∈᏿k

M andf (x)=

iaixi∈R[x,x−1], deﬁne

f (x) w=ν∈᏿k

M , whereν( z):=

i

aiw(z+i)∀z∈Zk. (6.3)

With this action,᏿kM becomes anR[x,x−1]-module. For subsetsI⊂R[x,x−1]andY ⊂

᏿k

M , consider

An_᏿k M (I)=

w∈᏿k

M |g w=0 for everyg∈I

,

AnR[x,x−1](Y )=

h∈R x,x−1_|_h_ν₌_{0 for every}_ν_∈_Y_. (6.4)

Obviously, An_᏿k M (I)⊂

᏿kM is anR[x,x−1]-submodule and AnR[x,x−1](Y ) R[x,x−1]is an ideal.

Definition _6.1. _Let _M _{be an}_R_{-module. We call} _w∈᏿_Mk _a _{linearly recursive} k-bisequence(resp., alinearly birecursivek-bisequence) if AnR[x](w)is a monic ideal (resp.,

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7. Reversible sequences over modules

7.1. LetM be anR-module. Ak-bisequenceu is said to be areverseofu∈᏿k M if

u|_Nk

0 =uand AnR[x](u) =AnR[x](u). A linearly recursivek-sequenceuwill be called reversibleifuhas a reverseu∈ᏸkM . With᏾kM ⊂ᏸkM , we denote theR[x]-submodule of reversiblek-sequences overM.

Lemma7.1(cf. [16, Proposition 14.11]). LetRbeArtinian.

(1)Every monic idealI R[x]contains a subset of monic polynomials

x_jdjqjxj|qjxjis reversible forj=1, . . . , k. (7.1)

(2)LetMbe anR-module. Then every linearly recursivek-bisequence overMis linearly birecursive (i.e.,ᏮkM =ᏸkM ).

Proof. _{(1) By [}₅_{, Theorem 8.7] every commutative Artinian ring is (up to} isomor-phism) a direct sum of local Artinian rings. Without loss of generality, letRbe a local Artinian ring. The Jacobson radical ofR,

J(R)= {r∈R|ris not invertible inR}, (7.2)

is nilpotent, hence there exists a positive integern such that J(R)n ₌_{0. Let}_I _{be a} monic ideal with a subset of monic polynomials{g1(x1), . . . , gk(xk)} ⊂I. Ifgj(xj)≡

fj(xj)(modJ(R)[xj])for j =1, . . . , k, thengj(xj)|fj(xj)n_{, where} _n_{is the index of}

nilpotency of the idealJ(R). Hence,fj(xj)n_∈_I_{. If we write}_fj(xj)n₌_xdj

j qj(xj)with (xj, qj(xj))=1, thenqj(0)∈U (R), that is,qj(xj)is a reversible polynomial forj= 1, . . . , k.

(2) Let u be a linearly recursivek-bisequence overM. IfR is Artinian, then by (1) AnR[x](u) contains a subset of monic polynomials{x

dj

j qj(xj)|qj(xj)is reversible for j=1, . . . , k}. Then for everyz∈Zk_{, we have}_(qj(xj)_u)(z

1, . . . , zj, . . . , zk)=(x

dj j qj(xj) u)(z 1, . . . , zj−dj, . . . , zk) = 0. Hence, {qj(xj) | i= 1, . . . , k} ⊂ AnR[x](u) , that is,

AnR[x](u) is a reversible ideal.

7.2. Backsolving. LetMbe anR-module. Letube a linearly recursive sequence over Mand assume that AnR[x](u)contains some monic polynomial of the formxd_q(x)₌ xd_(a

0+a1x+···+al−1xl−1+xl),a0∈U (R). Then

a0u(j+d)+a1u(j+d+1)+···+al−1u(j+d+l−1)+u(j+d+l)=0 ∀j≥0 (7.3)

and we get bybacksolvingauniquelinearly birecursive bisequence u∈An᏿M(q(x)) withu(n) =u(n)for alln≥d. In case l=0, The bisequenceu≡0 and is given for l≠0 by

u(z):=   

u(z), z≥d,

−a−01

a1u(z +1)+···+al−1u(z +l−1)+u(z +l)

, z < d. (7.4)

If there are two bisequencesv, w∈An_᏿

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AnR[x](u) =AnR[x](u). It is obvious that AnR[x](u) ⊆AnR[x](u). On the other hand, assume thatg(x)=m

j=0bjxj∈AnR[x](u). We prove by induction that(g u)(z) =0

for allz∈Z. First of all, note that for allz≥d, we have(g u)(z) =(g u)(z)=0. Now, letz0< dand assume that(g u)(z) =0 forz∈ {z0, z0+1, . . . , z0+l−1} ⊆Z.

Then we have, forz=z0−1,

g uz0−1

=

m

j=0

bjuj+z0−1 = m j=0 bj  l

i=1 −a−1

0 aiu

j+z0−1+i   = − l i=1

a−1 0 ai

m

j=0

bjuj+z0−1+i

= −

l

i=1

a−1 0 ai

g uz0−1+i=0.

(7.5)

Ifuis a linearly recursivek-sequence overM withk >1 and AnR[x](u)contains a

set of monic polynomials{xjdjqj(xj)|qjis reversible forj=1, . . . , k}, then we get by backsolvingthroughqj(xj)along thejth row forj=1, . . . , ka unique linearly birecur-sivek-bisequenceu∈An_᏿k

M (q1(x1), . . . , qk(xk))withu( n)=u(n)for alln≥dand it follows moreover that AnR[x](u) =AnR[x](u).

Lemma7.2. LetMbe anR-module.

(1)Every birecursivek-sequence overM is reversible with unique reverse (which we denote byRev(u)). Moreover,ᏮkM becomes a structure of anR[x,x−1]-module through f u:=(f Rev(u))|_Nk

0.

(2)IfRis Artinian, then every reversiblek-sequence overMis birecursive as well (i.e., Ꮾk

M =᏾ k M ).

Proof. (1) If u∈ Ꮾ_Mk, then An_R[_x](u) contains a set of reversible polynomials {qj(xj)|j=1, . . . , k}and we get by backsolving (seeSection 7.2) auniquelinearly bire-cursivek-bisequenceu ∈An

᏿Mk(q1(x1), . . . , qk(xk))with u( n)=u(n)for alln∈N k

0.

For the bisequenceu, we have as shown above AnR[x](u) =AnR[x](u), that is, u is a

reverse ofu. The last statement is obvious. (2) By (1),Ꮾk

M ⊆᏾ k

M . IfRis Artinian andu∈᏾ k

M with reverseu, then AnR[x](u)=

AnR[x](u) is, byLemma 7.1(2), reversible, that is,u∈ᏮMk.

Example7.3. The Fibonacci sequence=(0,1,1,2,3,5, . . .)has elementary charac-teristic polynomialf (x)=x2₋_x₋_{1. Since}_{f (}₀₎_{= −}_{1 is invertible in}_Z_{, we conclude}

thatis reversible with reverse

Rev()(z)=  



(z), z≥0,

[image:14.468.126.349.146.272.2]

Rev()(z+2)−Rev()(z+1), z <0. (7.6)

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Table7.1

z −4 −3 −2 −1 0 1 2 3 4

Rev()(z) −3 2 −1 1 0 1 1 2 3

Lemma7.4. There is an isomorphism ofR[x,x−1]-modules

Ꮾk M Ꮾ

k

M . (7.7)

Proof. ByLemma 7.2, we have the well-deﬁnedR[x,x−1]-linear map

Rev(−):ᏮMk →Ꮾ k

M , u→Rev(u), (7.8)

where Rev(u)is the linearly birecursive sequence deﬁned in (7.4). It is easy to see that Rev(−)is bijective with inverseuu|_Nk

0.

7.3. LetM be anR-module. We call a k-sequenceu∈᏿Mk periodic (resp., degen-erating) if xd₍_xt_u)₌_{0 for some}_d_∈_Nk

0 and t∈Nk (resp.,xd u=0 for some d∈Nk

0). It is clear that the subsetsᏼ

k M ⊆ᏸ

k

M of periodick-sequences andᏰ k M ⊆ᏸ

k M of degeneratingk-sequences areR[x]-submodules.

Remark_7.5 _{(see [}₁₆_{, Proposition 5.2])}. _If_M_{is a}_ﬁnite_R_{-module, then every linearly} recursive sequence overMis periodic (i.e.,ᏼM1=ᏸ

1 M ).

Proposition _7.6_{(see [}₁₆_{, Proposition 5.27])}. _Let_R _{be an} _arbitrary_commutative ring, M anR-module, and denote by ᏾ᏼk

M the set of reversible periodick-sequences overM. Then there is an isomorphism ofR[x]-modules

ᏼkM ᏰkM ⊕᏾ᏼkM . (7.9)

The following result generalizes Proposition 7.6 and describes the R[x]-module structure of arbitrary linearly recursivek-sequences ofR-modules, whereRis an Ar-tiniancommutative ground ring.

Proposition_7.7. _Let_M_{be an}_{R-module. If}_R_{is Artinian, then there are isomorphisms} ofR[x]-modules

ᏸk M Ᏸ

k M ⊕ᏸ

k M =Ᏸ

k M ⊕Ꮾ

k M Ᏸ

k M ⊕Ꮾ

k M =Ᏸ

k M ⊕᏾

k

M . (7.10)

Proof. If uis a linearly recursive sequence over M, then An_R[_x](u)contains, by Lemma 7.1(1), a set of monic polynomials {xdj_qj(xj)_|_qj(xj) _{is reversible for} _j ₌ 1, . . . , k}. Bybacksolving(see Section 7.2), we have a well-deﬁned morphism ofR[x] -modules

γ:ᏸkM →ᏸkM , u→u, (7.11)

whereuis theuniquelinearly birecursive bisequenceu∈An᏿M(q1, . . . , qk)withu( n)= u(n)for alln≥d. It is clear that Ke(γ)=Ᏸk

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ofR[x]-modules

β=ᏸk M →ᏸ

k

M , w→w|_Nk 0

. (7.12)

It is obvious thatγ◦β=id_ᏸk

M , hence the exact sequence

0 →ᏰkM →ᏸkM γ

→ᏸkM →0, (7.13)

ofR[x]-modules splits, that is,ᏸkM ᏰkM ⊕ᏸkM . SinceRis Artinian, we have by Lem-mata7.1(2) and7.2(2)ᏸk

M =Ꮾ k M andᏮ

k M =᏾

k

M . We are done now by the isomorphism ofR[x]-modulesᏮMkᏮ

k

M (Lemma 7.4).

7.4. The HopfR-algebraR[x,x−1_]_. _{Consider the commutative group}_G_{generated by} {xj|j=1, . . . , k}. Then the ring of Laurent polynomials R[x,x−1_]₌_RG_{has the}

struc-ture of acommutative cocommutativeHopfR-algebra(R[x,x−1_{], µ, η,}_∆_{, ε, S)}_{, where}_µ

(resp.,η) is the usual multiplication (resp., the usual unity) and for allz∈Z, j=1, . . . , k,

∆:R x,x−1_→_R _x_,_x−1_⊗

RR x,x−1, xjz→xzj⊗xjz, ε:R x,x−1 _→_R, _xz

j→1R,

S:R x,x−1 →R[x,x−1], x_jz→x−_jz.

(7.14)

Proposition7.8. LetRbe an arbitrary commutative ring. ThenR[x,x−1]is an ad-missible HopfR-algebra andR[x,x−1_]◦_{is a Hopf}_R-algebra.

Proof. Notice thatR[x,x−1]is a coﬁnitary HopfR-algebra byLemma 4.6(2). Con-sider the proof ofProposition 5.2and replaceR[x]withR[x,x−1_]_{. Then the map}

Tj:B →B, b→∆xjb (7.15)

is invertible with inverse

Tj:B →B, b→∆¯

x_j−1b. (7.16)

Then the matrixMjofTjis invertible andχj(0)∈U (R)forj=1, . . . , k. Consequently, ᏷R[x,x−1]satisﬁes axiom (A1). SinceR[x,x−1]/Ke(ε)R,᏷R[x,x−1]satisﬁes axiom (A2). Consider thebijectiveantipode S ofR[x,x−1_]_{. For every ideal}_{I R[}_x_,_x−1_]_,_S−1_(I)

R[x,x−1_] _{is an ideal and we have an isomorphism of}_R_-modules_R[_x_,_x−1_]/S−1_(I)

R[x,x−1_]/I_{. Hence,}_᏷

R[x,x−1]satisﬁes axiom (A3). Consequently,R[x,x−1]is an admis-sible HopfR-algebra. The last statement follows now byProposition 4.3.

For everyR-moduleM, we have an isomorphism ofR[x,x−1_]_-modules

ΨM:M x,x−1∗ →᏿Mk∗, ϕ→ z→ m→ϕ

mxz _(7.17)

with inverseu[mxz_u( _z_)(m)]_.

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Proposition7.9. LetRbe an arbitrary ring andManR-module. Then (7.17) induces an isomorphism ofR[x,x−1_]-modules

M x,x−1◦ᏮkM∗. (7.18)

Proof. _{Consider the isomorphism of} _R[_x_,_x−1_]_-modules _M[_x_,_x−1_]∗ ΨM ᏿k_M∗_{, see} (7.17). Let∈M[x,x−1_]◦_{. Then}_I₌_{0 for some}_R_-coﬁnite_R[_x_,_x−1_]_-ideal_{I R[}_x_,_x−1_]

and so I ΨM()=ΨM(I )=0. ByLemma 4.6(2), I is a reversible ideal and so AnR[x](u)⊃I∩R[x]is a reversible ideal, that is,ΨM()is linearly birecursive.

On the other hand, letu∈ᏮkM∗. Then AnR[x](u) is, by deﬁnition, a reversible ideal,

that is, it contains a subset of reversible polynomials{qj(xj),j=1, . . . , k}. Note that for arbitraryg∈R[x,x−1_]_{, we have}_gqj_Ψ−1

M (u) =ΨM−1(gqju) =ΨM−1(g (qju)) = 0 forj=1, . . . , k. ByLemma 4.6(2), the reversible ideal(q1(x1), . . . , qk(xk)) R[x,x−1]

isR-coﬁnite, that is,Ψ−1

M (u) ∈M[x,x−1]◦.

7.5. The HopfR-algebra structures onᏮk_and_Ꮾk_. _Let_R_{be an arbitrary ring and} consider the HopfR-algebra R[x,x−1_]_{. Then}_᏿k_RZk

R[x,x−1_]∗ _{is an}_R_-algebra with theHadamard product

∗:᏿k_⊗

R᏿k →᏿k, u⊗v→ z→u( z)v( z), (7.19)

and the unity

η:R →᏿k_, ₁

R→ z→1R

for everyz∈Zk_. _(7.20)

ByProposition 7.8,R[x,x−1_]◦_{is a Hopf}_R_{-algebra. So}_Ꮾk_R[_x_,_x−1_]◦_{inherits the} struc-ture of a HopfR-algebra(Ꮾk_,_∗_{g, ηg,}_∆

Ꮾk, ε_Ꮾk, S_Ꮾk), where∗gis the Hadamard prod-uct (5.14),ηgis the unity (5.15), and

∆Ꮾk:Ꮾk →Ꮾk⊗RᏮk, u→

t≤l−1

xt_u_⊗_eF t,

ε_Ꮾk:Ꮾk →R, u→u(0),

S_Ꮾk:Ꮾk →Ꮾk, u→ n→Rev(u)(−n).

(7.21)

Moreover,Ꮾk_R[_x_,_x−1_]◦_{becomes a Hopf}_R_-algebra₍_Ꮾk_,_∗_g_,_ηg_,_∆

Ꮾk,εᏮk,SᏮk), where∗is the Hadamard product (7.19),ηis the unity (7.20), and

∆Ꮾk:Ꮾk →Ꮾk⊗RᏮk, u→

t≤l−1

Rev !

xt_u

|_Nk 0

"

⊗ReveF t

,

εᏮk:Ꮾk →R, u→u( 0),

SᏮk:Ꮾk →Ꮾk, u→ z→u( −z).

(7.22)

Note that with these structures the isomorphismᏮk_Ꮾk_of_{Lemma 7.4}_{turns to be} an isomorphism of HopfR-algebras.

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Theorem7.10. IfRis Artinian, then there are isomorphisms ofR-bialgebras

ᏸk_Ᏸk_⊕_ᏸk₌_Ᏸk_⊕_Ꮾk_Ᏸk_⊕_Ꮾk₌_Ᏸk_⊕_᏾k_. _(7.23)

Proof. Consider the isomorphismᏸk Ᏸk⊕ᏸk, see (7.10). With the help of Lemmata4.5and7.1, one can show, as in [17, page 123], thatγ:ᏸk_→_ᏸk_{, see (}_7.11_), and β:ᏸk→ᏸk_{, see (}_7.12_{), are in fact bialgebra morphisms. Obviously, Ke}_(γ)₌ Ᏸk_⊂_ᏸk_{is an}_ᏸk_{-subbialgebra and we are done.}

As an analog toCorollary 5.8, we get the following corollary.

Corollary 7.11. LetM be an R[x,x−1]-module. Then there are isomorphisms of R[x,x−1_]◦_-comodules

ᏸk

M∗M x,x−

1◦_M∗_⊗

RR x,x−1

◦_M∗_⊗

RᏸRk. (7.24)

In particular,M[x,x−1_]◦₍_ᏸk

M∗) is a cofreeR[x,x−1]◦-comodule (ᏸ k

R -comodule).

As a consequence of [2, Theorem 2.4.7] and [2, Corollaray 2.5.10], we get the following corollary.

Corollary_7.12. _Let_R_{be Noetherian and consider the}_R-bialgebra_R[_x_;_g]◦_(resp., the HopfR-algebraR[x;p]◦, the HopfR-algebraR[x,x−1_]◦_{). If}_A_{is an}_{α-algebra (resp.,} anα-bialgebra, a Hopfα-algebra), then there are isomorphisms ofR-coalgebras (resp., R-bialgebras, HopfR-algebras)

A[x;g]◦A◦⊗RR[x;g]◦, A[x;p]◦A◦⊗RR[x;p]◦,

A x,x−1◦_A◦_⊗

RR x,x−1

◦_. (7.25)

7.6. Representative functions. Let G be a monoid (a group) and consider the R -algebraB=RG_{with pointwise multiplication. Then}_B_{is an}_RG_{-bimodule under the left} and right actions

(yf )(x)=f (xy), (f y)(x)=f (yx) ∀x, y∈G. (7.26)

We callf∈RG _an_{R-valued representative function}_{on the monoid}_G_if_{(RG)f (RG)}_is ﬁnitely generated as anR-module. IfR is Noetherian, then the subset᏾(G)⊂RG _of all representative functions onGis anRG-subbimodule. Moreover, we deduce from [4, Theorem 2.13 and Corollary 2.15] that in case(RG)◦⊂RG_{is pure, we have an} isomor-phism ofR-bialgebras (HopfR-algebras)᏾(G)(RG)◦.

Corollary7.13. LetRbe Noetherian. (1)Considering the monoid(Nk

0,+), there are isomorphisms ofR-bialgebras

᏾Nk

0

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(2)Considering the group(Zk_,₊_{), there are isomorphisms of Hopf}_R-algebras

᏾Zk_R _x_,_x−1◦_Ꮾk R Ꮾ

k

R . (7.28)

Acknowledgments. I am so grateful to my advisor Professor Robert Wisbauer for his wonderful supervision and continuous encouragement and support. The author is also grateful to the referees for helpful suggestions and for drawing his attention to [12,13] by Kurakin.

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Jawad Y. Abuhlail: Mathematics Department, Birzeit University, P.O. Box 14, Birzeit, West Bank, Palestine

Current address: Box # 281 KFUPM, 31261 Dhaharan, Saudia Arabia