On Linear Difference Equations over Rings and Modules

In this note 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 commutative ground rings.


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 importance mainly because of recent applications in coding theory and cryptography (e.g. [HN99], [KKMMN99]).
In this note we develop a coalgebraic aspect to the study of solutions of linear difference equations over arbitrary rings and modules. For some of our results we assume that the ground ring is artinian. Our results generalize also previous results of us in [AG-TW00] and [Abu01,Kapitel 4]. A standard reference for the theory of linearly recursive sequences over rings and modules is the comprehensive work of A. Mikhalev et al. [KKMN95]. For the theory of Hopf algebras the reader may refer to any of the classical references (e.g. [Swe69], [Abe80] and [Mon93]).
With R we denote a commutative ring with 1 R = 0 R and with U(R) = {r ∈ R| r is invertible} the group of units of R. The category of R-(bi)modules will be denoted by M R . For an R-module M, we call an R-submodule K ⊂ M pure (in the sense of Cohn), if for every R-module N the induced map ι k ⊗ id N : For an R-algebra A and an A-module M, we call an A-submodule K ⊂ M R-cofinite, if M/K is f.g. in M R . For an R-algebra A we denote by K A the class of R-cofinite ideals. If A is an R-algebra with K A a filter, then we define for every left A-module M the finite dual right A-module M • := {f ∈ M * | Ke(f ) ⊃ IM for some A-ideal I with A/I f.g.}. (1) With N resp. Z we denote the set of natural numbers resp. the ring of integers. Moreover we set N 0 := {0, 1, 2, 3, ...}. For an n × n matrix M over R we denote the characteristic polynomial with χ(M). The identity matrix of order n over R is denoted by E n . For an m × n matrix A and a k × l matrix B, the Kronecker product (tensor product) of A and B is the mk × nl matrix
1.1. Let M be an R-module, l =(l 1 , ..., l k ) ∈ N k 0 and consider the system of linear difference equations (ab. SLDE) x n+(0,l 2 ,0,...,0) + where the p jl 's are R-valued functions and the g j 's are M-valued functions defined for all n ∈ N k 0 . If the g j 's are identically zero, then (3) is said to be a homogenous SLDE. If the p jl 's are constants, then (3) is said to be a SLDE with constant coefficients.

1.2.
For an R-module M and k ≥ 1 let Note that An <k> , the companion matrix of f is defined to be the l × l matrix S f is a matrix that has f (x) as its characteristic polynomial as well as its minimum polynomial ([Jon73, Theorem 4.18]). Definition 1.4. An ideal I ⊳ R[x] will be called monic, if it contains a non-empty subset of monic polynomials In this case the polynomials (6) are called elementary polynomials and (f 1 ( will be called reversible, if it contains a subset of reversible polynomials {q 1 (x 1 ), ..., q k (x k )}.
1.5. Let M be an R-module. We call u ∈ S <k> M a linearly recursive k-sequence (resp. a linearly birecursive k-sequence), if An R[x] (u) is a monic ideal (resp. a reversible ideal). Note that a k-sequence u ∈ S <k> is monic for j = 1, ..., n and write for every n = (n 1 , ..., Consequently ω is completely determined by the initial polyhedron of values ω(Π F ). For

Examples
We give now some examples of linearly recursive sequences.
For more examples the reader may refer to [KKMN95].
Example 1.7. (Geometric progression). Let M be an R-module, m ∈ M, r ∈ R and consider w ∈ S M given by w(n) := r n m for every n ∈ N 0 .
Then w ∈ L M with initial condition w(0) = m and elementary characteristic polynomial Then w ∈ L M with initial vector (p, p + q) and elementary characteristic polynomial Remark 1.9. An example of a non linearly recursive sequences over Z is the sequence of prime positive numbers {2, 3, 5, 7, ...}.
be a subset of monic polynomials.

Admissible R-bialgebras and Hopf R-algebras
For every R-coalgebra (C, ∆ C , ε C ) there is a dual R-algebra C * := Hom R (C, R) with multiplication the so called convolution product and unity ε C . Although every algebra A has a dual coalgebra, if the ground ring is hereditary noetherian (e.g. a field), the existence of dual coalgebras of algebras over an arbitrary commutative ground rings is not guaranteed!! One way to handle this problem is to restrict the class of R-algebras, for which the dual R-coalgebras are defined.
Definition 1.11. Let A be an R-algebra (resp. an R-bialgebra, a Hopf R-algebra). Then we call A : 1. an α-algebra (resp. an α-bialgebra, a Hopf α-algebra), if K A is a filter and A • ⊂ R A is pure.
2. cofinitary, if K A is a filter and for every I ∈ K A there exists an A-ideal I ⊆ I with A/I f.g. and projective.
1.12. Let H be an R-bialgebra and consider the class of R-cofinite H-ideals K H . We call H an admissible R-bialgebra, if H is cofinitary and K H satisfies the following axioms: and We call a Hopf R-algebra H an admissible Hopf R-algebra, if H is cofinitary, K H satisfies (A1), (A2) and Remark 1.13. It follows from the proof of [AG-TL01, Proposition 4.2.], that every cofinitary R-algebra (resp. R-bialgebra, Hopf R-algebra) is an α-algebra (resp. an α-bialgebra, a Hopf α-algebra). By ([Abu01, Lemma 2.5.6.]) every cofinitary bialgebra (Hopf algebra) over a noetherian ground ring is admissible. 1. If A is a cofinitary R-algebra, then A • is an R-coalgebra. If H is an admissible Rbialgebra (resp. an admissible Hopf R-algebra), then H • is an R-bialgebra (resp. a Hopf R-algebra).
Proof.  2 Linearly (bi)recursive sequences In this section we study the linearly (bi )recursive k-sequences over R-modules, where R is an arbitrary commutative ground ring.
Remarks 2.3. 1. Let R be an integral domain, then it follows by [Grü69, Theorem 1.3.6.] that for every set G, the class of group-like elements of the R-coalgebra RG is G itself. Then one can show as in the field case [CG93], that R[x; g] and R[x; p] are the only possible R-bialgebra structures on R[x] with the usual multiplication and the usual unity.

The R-bialgebra R[x; g] has no antipode, because the group-like elements in a Hopf
R-algebra should be invertible.
Proposition 2.5. Let M be an R-module. Then (11) induces an isomorphism of 2.6. The coalgebra structure on L <k> .
2.9. The Hopf R-algebra (L <k> R ; p). Consider the Hopf R-algebra R[x; p]. Then S <k> ≃ R N k 0 ≃ R[x; p] * is an R-algebra with multiplication given by the Hurwitz product * p : and the unity η p : R → S <k> , 1 R → [n → δ n,0 ] for every n ∈ N k 0 .
By Propositions 2.4 and 2.5 (L <k> R ; p) ≃ R[x; p] • has the structure of a Hopf R-algebra with the coalgebra structure described in 2.6, the Hurwitz product (17), the unity (18) and the antipode Proposition 2.10. ([Kur02, Theorem 3]) Let u and v be linearly recursive sequences over R of orders m, n and with characteristic polynomials f (x), g(x) respectively. Then 1. u ⋆ g v is a linearly recursive sequence over R of order m · n and characteristic polynomial χ(S f ⊗ S g ); 2. u ⋆ p v is a linearly recursive sequence over R of order m · n and characteristic polynomial χ(S f ⊗ E n + E m ⊗ S g ).
Then {x n } ∞ n=0 is a linearly recursive sequence over R with characteristic polynomial f (x) = x 3 − x 2 + x − 1 and {y n } ∞ n=0 is a linearly recursive sequence over R with characteristic polynomial g(x) = x 2 − x + 1.
as right C-comodules (this is one reason of the terminology cofree).
As a direct consequence of Lemma 1.17 we get Corollary 2.14. Let M be an R[x]-module. Then we have an isomorphism of

Linearly (bi)recursive bisequences
In this section we consider the linearly (bi )recursive k-bisequences and the reversible ksequences over R-modules, where R is an arbitrary commutative ground ring. We generalize results of [LT90] and [KKMN95] concerning the bialgebra structure of the linearly recursive sequences over a base field to the case of arbitrary artinian ground rings.
1. By [AM69,8.7] every commutative artinian ring is (up to isomorphism) a direct sum of local artinian rings. W.l.o.g. let R be a local artinian ring. The Jacobson radical of R J(R) = {r ∈ R| r is not invertible in R} is nilpotent, hence there exists a positive integer n, such that J(R) n = 0. Let I be a monic ideal with a subset of monic polynomials with (x j , q j (x j )) = 1, then q j (0) ∈ U(R), i.e. q j (x j ) is a reversible polynomial for j = 1, ..., k.
2. Let u be a linearly recursive k-bisequence over M. If R is artinian, then An R[x] ( u) contains by (1) a subset of monic polynomials {x 3.6. Backsolving. Let M be an R-module. Let u be a linearly recursive sequence over M and assume that An R[x] (u) contains some monic polynomial of the form x d q(x) = x d (a 0 + a 1 x + ... + a l−1 x l−1 + x l ), a 0 ∈ U(R). Then a 0 u(j + d) + a 1 u(j + d + 1) + ... + a l−1 u(j + d + l − 1) + u(j + d + l) = 0 for all j ≥ 0 and we get by Backsolving a unique linearly birecursive bisequence u ∈ An S M (q(x)) with u(n) = u(n) for all n ≥ d. The bisequence u ≡ 0 in case l = 0 and is given for l = 0 by 0 (a 1 u(z + 1) + ... + a l−1 u(z + l − 1) + u(z + l)), z < d.
If there are two bisequences v, w ∈ An S M (q(x)) with v(n) = u(n) = w(n) for all n ≥ d, then one can easily show by backsolving using q(x) that v = w. b j x j ∈ An R[x] (u). We prove by induction that (g ⇀ u)(z) = 0 for all z ∈ Z.
First of all, note that for all z ≥ d we have (g ⇀ u)(z) = (g ⇀ u)(z) = 0. Now let z 0 < d and assume that (g ⇀ u)(z) = 0 for z ∈ {z 0 , z 0 + 1, ..., z 0 + l − 1} ⊆ Z. Then we have for z = z 0 − 1 : If u is a linearly recursive k-sequence over M with k > 1 and An R[x] (u) contains a set of monic polynomials {x d j j q j (x j ) | q j is reversible for j = 1, ..., k}, then we get by backsolving through q j (x j ) along the j-th row for j = 1, ..., k a unique linearly birecursive k-bisequence u ∈ An S <k> 3.17. The Hopf R-algebra structures on B <k> and B <k> . Let R be an arbitrary ring and consider the Hopf R-algebra R[x, and the unity η : where ⋆ g is the Hadamard product (15), η g is the unity (16) and where ⋆ is the Hadamard product (29), η is the unity (30) and Note that with these structures the isomorphism B <k> ≃ B <k> of Lemma 3.9 turns to be an isomorphism of Hopf R-algebras.
The following theorem extends the corresponding result from the case of a base field [LT90, Page 124] (see also [KKMN95,14.15]) to the case of arbitrary artinian ground rings: Theorem 3.18. If R is artinian, then there are isomorphisms of R-bialgebras L <k> ≃ D <k> ⊕ L <k> = D <k> ⊕ B <k> ≃ D <k> ⊕ B <k> = D <k> ⊕ R <k> .
As an analog to Corollary (2.14) we get Corollary 3.19. Let M be an R[x, x −1 ]-module. Then we have an isomorphism of R[x, Corollary 3.20. Let R be noetherian and consider the R-bialgebra R[x; g] • (resp. the Hopf R-algebra R[x; p] • , the Hopf R-algebra R[x, x −1 ] • ). If A is an α-algebra (resp. an α-bialgebra, a Hopf α-algebra), then we have isomorphism of R-coalgebras (resp. Rbialgebras, Hopf R-algebras) 3.21. Representative functions. Let G be a monoid (a group) and consider the Ralgebra B = R G with pointwise multiplication. Then B is an RG-bimodule under the left and right actions (yf )(x) = f (xy) and (f y)(x) = f (yx) for all x, y ∈ G.
We call f ∈ R G an R-valued representative function on the monoid G, if (RG)f (RG) is finitely generated as an R-module. If R is noetherian, then the subset R(G) ⊂ R G of all representative functions on G is an RG-subbimodule. Moreover we deduce from [AG-TW00, Theorem 2.13, Corollary 2.15] that in case (RG) • ⊂ R G is pure, we have an isomorphism of R-bialgebras (Hopf R-algebras) R(G) ≃ (RG) • .
2. Considering the monoid (Z k , +) we have isomorphisms of Hopf R-algebras