FINITE NONASSOCIATIVE ALGEBRAS OBTAINED FROM SKEW POLYNOMIALS AND POSSIBLE APPLICATIONS TO ( f, σ, δ ) -CODES

. Let S be a unital ring, S [ t ; σ,δ ] a skew polynomial ring where σ is an injective endomorphism and δ a left σ -derivation, and suppose f ∈ S [ t ; σ,δ ] has degree m and an invertible leading coeﬃcient. Using right division by f to deﬁne the multiplication, we obtain unital nonassociative algebras S f on the set of skew polynomials in S [ t ; σ,δ ] of degree less than m . We study the structure of these algebras. When S is a Galois ring and f base irreducible, these algebras yield families of ﬁnite unital nonassociative rings A , whose set of (left or right) zero divisors has the form pA for some prime p . For reducible f , the S f can be employed both to design linear ( f,σ,δ )-codes over unital rings and to study their behaviour.


Introduction
Let S be a unital ring. In the present paper we construct a new class of nonassociative unital rings out of subsets of the skew polynomial ring R = S[t; σ, δ] where σ is an injective endomorphism and δ a left σ-derivation. Given a polynomial f ∈ R = S[t; σ, δ] of degree m, whose leading coefficient is a unit, it is well-known by now (e.g., cf. [31], [26], [13] for commutative S) that it is possible to define a right division by f : for all g(t) ∈ R, there exist uniquely determined r(t), q(t) ∈ R with deg(r) < m, such that g(t) = q(t)f (t) + r(t). What is much less known is the fact that we can take the additive group {g ∈ R | deg(g) < m} of skew polynomials of degree less than m, i.e. the canonical representatives of the remainders in R of right division by f , and define a nonassociative unital ring structure • on it via g • h = gh mod r f . The resulting nonassociative ring S f , also denoted S[t; σ, δ]/S[t; σ, δ]f , is a unital nonassociative algebra over a commutative subring of S. If f is two-sided (also called invariant), i.e. if S[t; σ, δ]f is a two-sided ideal, then S[t; σ, δ]/S[t; σ, δ]f is the well-known associative quotient algebra obtained by factoring out a principal two-sided ideal. This generalizes a construction introduced by Petit for the case when S is a division ring and thus R = S[t; σ, δ] left and right Euclidean [33].
The algebras S f were previously introduced by Petit, but only for the case that S is a division ring, hence S[t; σ, δ] left and right Euclidean [33]. In that setting, they already appeared in [12], [13], [32], and were used in space-time block coding, cf. [47], [40], [39].
We present two possible applications: We first use our algebras to construct new families of finite nonassociative unital rings, especially generalized nonassociative Galois rings. Generalized nonassociative Galois rings were introduced in [21] and investigated in [23], [22], [24]. They are expected to have wide-ranging applications in coding theory and cryptography [21].
As a second application, we point out the canonical connection between the algebras S f and cyclic (f, σ, δ)-codes. This connection was first mentioned in [34] for S being a division ring. Well-known results from the literature, e.g. on the pseudo-linear map T f [11] and on polynomials in Ore extensions from [8] or [29], are rephrased in this setting and put into a nonassociative context.
The paper is organized as follows. We establish our basic terminology in Section 2, define the algebras S f in Section 3 and investigate their basic structure in Section 4.
The matrix representing left multiplication with t in S f yields the pseudolinear transformation T f associated to f defined in [8] which is discussed in Section 5. We generalize [28,Theorem 13 (2), (3), (4)] and show that if S f has no zero divisors then T f is irreducible, i.e. {0} and S m are the only T f -invariant left S-submodules of S m .
In Section 6, we assume that S is a finite chain ring. If f is base irreducible then S f is a generalized nonassociative Galois ring. This yields new families of generalized nonassociative Galois rings.
We consider the connection between the algebras S f and cyclic (f, σ, δ)-codes, in particular skew-constacyclic codes over finite chain rings, in Section 7: We rephrase some results (for instance from [4], [7], [8], [26], [6]), by employing the algebras S f instead of dealing with cosets in the quotient module S[t; σ, δ]/S[t; σ, δ]f . For instance, the matrix generating a cyclic (f, σ, δ)-code C ⊂ S m represents the right multiplication R g in S f , calculated with respect to the basis 1, t, . . . , t m−1 , identifying an element h = m−1 i=0 a i t i with the vector (a 0 , . . . , a m−1 ), cf. [8]. This matrix generalizes the circulant matrix from [15] and is a control matrix of C. We also show how to obtain semi-multiplicative maps using their right multiplication. This paper is the starting point for several applications of the algebras S f to coding theory, e.g. to coset coding, and related areas. Some are briefly explained in Section 7.

Preliminaries
2.1. Nonassociative algebras. Let R be a unital commutative ring and let A be an R-module. We call A an algebra over R if there exists an R-bilinear map A × A → A, (x, y) → x · y, denoted simply by juxtaposition xy, the multiplication of A. An algebra A is called unital if there is an element in A, denoted by 1, such that 1x = x1 = x for all x ∈ A. We will only consider unital algebras.
For an R-algebra A, associativity in A is measured by the associator is an associative subalgebra of A containing R1 and x(yz) = (xy)z whenever one of the elements x, y, z is in Nuc(A). The commuter of A is defined as Comm(A) = {x ∈ A | xy = yx for all y ∈ A} and the center of A is C(A) = {x ∈ A | x ∈ Nuc(A) and xy = yx for all y ∈ A} [43].
An algebra A = 0 over a field F is called a division algebra if for any a ∈ A, a = 0, the left multiplication with a, L a (x) = ax, and the right multiplication with a, R a (x) = xa, are bijective. A division algebra A does not have zero divisors. If A is a finite-dimensional algebra over F , then A is a division algebra over F if and only if A has no zero divisors.
For f = a 0 + a 1 t + · · · + a n t n with a n = 0 define deg(f ) = n and deg(0) = −∞. Then deg(gh) ≤ deg(g) + deg(h) (with equality if h has an invertible leading coefficient, or g has an invertible leading coefficient and σ is injective, or if S is a division ring). An element f ∈ R is irreducible in R if it is not a unit and it has no proper factors, i.e if there do not exist Suppose D is a division ring. Then R = D[t; σ, δ] is a left principal ideal domain (i.e., every left ideal in R is of the form Rf ) and there is a right division algorithm in R [25, p. 3]: for all g, f ∈ R, g = 0, there exist unique r, q ∈ R, and deg(r) < deg(f ), such that g = qf + r (cf. Jacobson [25] and Petit [33]

Nonassociative rings obtained from skew polynomials rings
From now on, let S be a unital ring and S[t; σ, δ] a skew polynomial ring where σ is injective. S[t; σ, δ] is generally neither a left nor a right Euclidean ring (unless S is a division ring). Nonetheless, we can still perform a left and right division by a polynomial f ∈ R = S[t; σ, δ], if f (t) = m i=0 d i t i has an invertible leading coefficient LC(f ) = d m (this was already observed for twisted polynomial rings and special cases of S and assuming σ ∈ Aut(S) for instance in [31, p. 391], [26, p. 4], [13, 3.1]): have degree m and an invertible leading coefficient.
(ii) Assume σ ∈ Aut(S). Then for all g(t) ∈ R of degree l ≥ m, there exist uniquely determined r(t), q(t) ∈ R with deg(r) < deg(f ), such that Note that we used that ∆ l−m,l−m (d m ) = σ l−m (d m ) in the last equation. Therefore the polynomial g(t) − s l σ l−m (d m )t l−m f (t) has degree < l. By iterating this argument, we find r, q ∈ R with deg(r) < deg(f ), such that To prove uniqueness of q(t) and the remainder r(t), suppose we have Then (q 1 (t) − q 2 (t))f (t) = r 2 (t) − r 1 (t). If q 1 (t) − q 2 (t) = 0 and observing that f has invertible leading coefficient such that σ(d m ) j cannot be a zero divisor for any positive j, we conclude that the degree of the left-hand side of the equation is greater than deg(f ) and the degree of r 2 (t) − r 1 (t) is less than deg(f ), thus q 1 (t) = q 2 (t) and r 1 (t) = r 2 (t).
(ii) The proof is along similar lines as the one of (i), using that the polynomial )t l−m has degree < l and iterating this argument. The uniqueness of q(t) and the remainder is proved analogously as in (i).
In the following, we always assume that f (t) ∈ S[t; σ, δ] has degree m > 1 and an invertible leading coefficient LC(f ).
Let mod r f denote the remainder of right division by f and mod l f the remainder of left division by f . Since the remainders are uniquely determined, the skew polynomials of degree less that m canonically represent the elements of the left S[t; σ, δ]-module S[t; σ, δ]/S[t; σ, δ]f and when σ ∈ Aut(S), for the right S[t; σ, δ]module S[t; σ, δ]/f S[t; σ, δ].
(i) The additive group {g ∈ R | deg(g) < m} together with the multiplication g • h = gh mod r f defined for all g, h ∈ R of degree less than m, is a unital nonassociative ring S f also denoted by R/Rf . (ii) Suppose σ ∈ Aut(S). Then the additive group {g ∈ R | deg(g) < m} together with the multiplication g h = gh mod l f defined for all g, h ∈ R of degree less than m, is a unital nonassociative ring f S also denoted by R/f R.
S f and f S are unital algebras over S 0 = {a ∈ S | ah = ha for all h ∈ S f }, which is a commutative subring of S. If S is a division ring, Definition 3.1 is Petit's algebra construction [33] and S 0 is a subfield of S. In the following, we therefore call the algebras S f Petit algebras. Remark 1. (i) Let g, h ∈ R have degrees less than m. If deg(gh) < m then the multiplication g • h in S f and g h in f S is the usual multiplication of polynomials in R.
(ii) If Rf is a two-sided ideal in R (i.e. f is two-sided, also called invariant) then S f is the associative quotient algebra obtained by factoring out the ideal generated by a two-sided f ∈ S[t; σ, δ].
(iii) If f ∈ S[t; σ, δ] is reducible then S f contains zero divisors: if f (t) = g(t)h(t) then g(t) and h(t) are zero divisors in S f . The argument leading up to [33, Section 2., (6)] shows that if S is a division ring, then S f has no zero divisors if and only if f is irreducible, which is in turn equivalent to S f being a right division ring (i.e., right multiplication R h in S f is bijective for all 0 = h ∈ S f ).
However, for general rings S it can happen that S f has zero divisors, even when f is irreducible.
(iv) For all invertible a ∈ S we have S f = S af , so that without loss of generality it suffices to only consider monic polynomials in the construction.
It suffices to consider the algebras S f , since we have the following canonical anti-automorphism (cf. [33, (1)] when S is a division ring, the proof is analogous): have an invertible leading coefficient and let σ ∈ Aut(S). The canonical anti-automorphism induces an anti-automorphism between the rings

Some structure theory
In the following, let f ∈ R = S[t; σ, δ] be monic of degree m > 1 and σ injective. When S is a division ring, the structure of S f is extensively investigated in [33]. For instance, if S is a division ring and the S 0 -algebra S f is finite-dimensional, or free of finite rank as a right module over its right nucleus, then S f is a division algebra if and only if f (t) is irreducible [33, (9)].
Some of the results in [33] carry over to our more general setting: {g ∈ R | deg(g) < m and f g ∈ Rf } = Nuc r (S f ). When S is a division ring, the inclusions become equalities. (iv) We have t ∈ Nuc r (S f ), if and only if the powers of t are associative, if and If the endomorphism L t which is the left multiplication by t as defined in Section 2.1 is surjective then σ is surjective. In particular, if S is a division ring and f irreducible, then L t surjective implies σ surjective.
Moreover, if σ is bijective then L t is surjective.
Thus t ∈ Nuc r (S f ) and also S ⊂ Nuc r (S f ). This means f (t)t ∈ Rf (t) and for all a ∈ S, also f (t)a = g(t)f (t) for a suitable g(t) ∈ R. Comparing degrees (recall we assume f to have an invertible leading coefficient) we see that g(t) = b ∈ S, so we get f (t)t ∈ Rf (t) and for all a ∈ S, also f (t)a = bf (t) for a suitable b ∈ R. Thus f is invariant, i.e. two-sided.
(iii) The proof of the first two inclusions and that {g ∈ R | deg(g) < m and f g ∈ Rf } ⊂ Nuc r (S f ) is similar to [33, (2)] (which proves the result for S being a division ring), as this inclusion does not need S to be a division ring. For instance, If a has degree 0 then p = 0 as observed in [33, (2) [33, (5)] shows that the other implications can be proved analogously as in [33, (5)], also when also holds when S is not a division algebra, since we still have that we have [t i , t j , t k ] = 0 for all i, j, k < m with i + j < m analogously as in [33,6].
Comparing the constants in this equation, we obtain that for all b ∈ S there is u m−1 ∈ S such that σ(u m−1 ) = bd 0 , i.e. for all c ∈ S there is u m−1 ∈ S such that σ(u m−1 ) = c [9].
The statement that if S is a division ring and f irreducible then L t is surjective implies σ surjective is [33, Section 2., (6)] and follows as a special case now.
If σ is bijective then L t is surjective: .
Recall that the largest subalgebra of R = S[t; σ, δ] in which Rf is a two-sided ideal is the idealizer The eigenring of f is then defined as the quotient E(f ) = I(f )/Rf . The eigenring E(f ) = {g ∈ R | degg < m and f g ∈ Rf } equals the right nucleus Nuc r (S f ) by Theorem 4.1 (iii) (or see [33, (2)] if S is a division algebra) which, as the right nucleus, is an associative subalgebra of S f , cf. Section 2.1. More precisely, the multiplication • in S f makes Nuc r (S f ) into an associative algebra which equals the associative quotient ring E(f ) equipped with the canonical multiplication induced on it by the multiplication on the ring I(f ) ⊂ R. When S is a division ring, non-trivial zero divisors in E(f ) = Nuc r (S f ) correspond to factors of f :
All non-zero left ideals in S f which contain a polynomial g of minimal degree with invertible leading coefficient are principal ideal generated by g, and g is a right divisor of f in R.
(ii) Each principal left ideal generated by a right divisor of f is an S-module which is isomorphic to a submodule of S m .
(iii) If f is irreducible, then S f has no non-trivial principal left ideals which contain a polynomial of minimal degree with invertible leading coefficient.
The proof is straightforward. If there is no polynomial g of minimal degree with invertible leading coefficient in a non-zero left ideal, then the ideal need not be principal, see [26,Theorem 4.1] for examples.
If t is left-invertible in S f and S a division ring, the two sets are equal.
If t is left-invertible and S a division ring, the two sets are equal.

4.1.
When S is an integral domain. In this section, let S be a commutative integral domain with quotient field K, f be monic and σ injective as before. Then σ and δ canonical extend to σ and δ to K via for all a, b ∈ S, b = 0.  [46] (to be precise, [46] looks at their opposite algebras). We generalize their definition (see [32] for the associative set-up): Let S/S 0 be an extension of commutative rings, σ ∈ Aut(S) and G = σ a finite cyclic group of order m acting on S such that the action is trivial on S 0 . For any c ∈ S, the generalized (associative or nonassociative) cyclic algebra A = (S/S 0 , σ, c) is the m-dimensional S-module A = S ⊕ St ⊕ St 2 ⊕ · · · ⊕ St m−1 where multiplication is given by the following relations for all a, b ∈ S, 0 ≤ i, j, < m, which then are extended linearly to all elements of A: If σ ∈ Aut(S), then (S/S 0 , σ, c) = S f for f (t) = t m − c ∈ S[t; σ] and S 0 = Fix(σ). If c ∈ S \ S 0 , the algebra (S/S 0 , σ, c) has nucleus S and center S 0 .
Suppose S 0 and S are integral domains with quotient fields F and K. Canonically extend σ to an automorphism σ : K → K, then if m is prime, (S/S 0 , σ, c) = S f has no zero divisors for any choice of c ∈ S \ S 0 (since then (K/F, σ, c) always is a nonassociative cyclic division algebra and contains S f ).
Generalized associative cyclic algebras are used in [13], generalized nonassociative cyclic algebras in [35]. Right multiplication with 0 = h ∈ S f in S f , R h : S f −→ S f , p → ph, is an S-module endomorphism [33]. After expressing R h in matrix form with respect to the S-basis 1, t, . . . , t m−1 of S f , the map induces an injective S-linear map

Pseudolinear maps
If we consider S f as a right Nuc r (S f )-module then L h is a Nuc r (S f )-module endomorphism.
For a two-sided f , γ is the right regular representation and λ is the left regular representation of the associative algebra S f .
If S is a commutative ring and det(γ(h)) = det Y = 0, then h is a right zero divisor in S f . Moreover, S f is a division algebra if and only if γ(h) is an invertible matrix for every nonzero h ∈ S f . [15], where S is a finite field and f (t) = t n − a, δ = 0, γ(h) = Y is the circulant matrix M θ a . (ii) If S is not commutative, but contains a suitable commutative subring, it is still possible to define a matrix representing left or right multiplication in the S 0 -algebra S f where the entries of the matrix lie in a commutative subring of S which strictly contains S 0 and which displays the same behaviour as above. This is a particularity of Petit's algebras, and not always possible for nonassociative algebras in general. It reflects the fact that the left nucleus of S f always contains S (and thus is rather 'large') and that also the right nucleus may contain S or subalgebras of S, depending on the f used in the construction.

Remark 4. (i) In
For instance, this is the case (and was used when designing fast-decodable spacetime block codes, e.g. in [40], [39], [37]) when S is a cyclic division algebra S = (K/F, ρ, c) of degree n and f (t) = t m − d ∈ S[t; σ], with σ suitably chosen. The m × m matrix γ(h) = Y consequently has its entries in (K/F, ρ, c). We can then substitute each entry in the matrix, which has the form σ i (x) for some x ∈ (K/F, ρ, c), perhaps timed with the scalar d, with an n × n matrix: take the matrix of the right regular representation of x over K in (K/F, ρ, c), apply σ i to each of its entries and using scalar multiplication by d if applicable. We obtain an mn × mn matrix X with entries in the field K, which still represents right multiplication with an element in S f , but now written with respect to the canonical K-basis 1, . . . , e, t, . . . , et, . . . , e n−1 t m−1 of S f , 1, e, . . . , e n−1 being the canonical basis of (K/F, ρ, c). Again det X = 0 implies that h is a zero divisor in S f , and were S f is a division algebra if and only if X is invertible for every non-zero h ∈ S f . The interested reader is referred to [47], [40], [39], [37] for the details which would be beyond the scope of this paper.
be the companion matrix of f . Then or equivalently by when calculated with respect to the basis 1, t, . . . , t m−1 , identifying an element h = n i=0 a i t i with the vector (a 1 , . . . , a n ). Proof. This is proved for instance in [28, Theorem 13 (2), (3), (4)] for δ = 0, f irreducible and S a finite field. The proofs generalize easily and mostly verbatim to our more general setting.
From Theorem 4.1 (vi) together with Theorem 5.1 (i) we obtain: If σ is not surjective then the pseudolinear transformation T f is not surjective. In particular, if S is a division ring, f irreducible and σ is not surjective then T f is not surjective.
Moreover, if σ is bijective then T f is surjective.
for all a ∈ S, h ∈ S m [8], so L t n is usually not (σ, δ)-pseudolinear anymore.
(ii) Right multiplication with h in S f induces the injective S-linear map f is two-sided is equivalent to γ being the right regular representation of S f . In that case, γ is an injective ring homomorphism. In particular, (1) and (3)  (iii) Suppose f = h g = gh. Right multiplication in S f induces the left S-module endomorphisms R h and R g . We have g ∈ ker(R h ) = {u ∈ R | deg(u) < m and uh ∈ Rf } and h ∈ ker(R g ) = {u ∈ R | deg(u) < m and ug ∈ Rf }, cf. [29,Lemma 3] or [15,Theorem 6.6]. If f is two-sided, ker(R g ) = S f h and ker(R h ) = S f g.
Left multiplication in S f induces the right S 0 -module endomorphisms L h and L g . We have g ∈ ker(L h ) = {u ∈ R | deg(u) < m and hu ∈ Rf } and h ∈ ker(L g ) = {u ∈ R | deg(u) < m and gu ∈ Rf }. If f is two-sided, ker(L h ) = gS f and ker(L g ) = h S f . Furthermore, (iii) and (iv) tie in with or generalize (4), (5) in [15,Theorem 6.6].
6. Finite nonassociative rings obtained from skew polynomials over finite chain-rings 6.1. Finite chain rings (cf. for instance [31]). When S is a finite ring, S f is a finite unital nonassociative ring with |S| m elements and a finite unital nonassociative algebra over the finite subring S 0 of S. E.g., if S is a finite field and f irreducible, then S f is a finite unital nonassociative division ring, also called a semifield [28].
Every ideal of a finite chain ring is principal and its maximal ideal is unique. In particular, R is a local ring and the residue field K = R/(γ), where γ is a generator of its maximal ideal m, is a finite field. The ideals (γ i ) = γ i R of R form the proper chain R = (1) ⊇ (γ) ⊇ (γ 2 ) ⊇ · · · ⊇ (γ e ) = (0). The integer e is called the nilpotency index of R. If K has q elements, then |R| = q e . If π : Let R and S be two finite chain rings such that R ⊂ S and 1 R = 1 S Then S is an extension of R denoted S/R. If m is the maximal ideal of R and M the one of S, then S/R is called separable if mS = M . The Galois group of S/R is the group G of all automorphisms of S which are the identity when restricted to R. A separable extension S/R is called Galois if S G = {s ∈ S | τ (s) = s for all τ ∈ G} = R. This is [48,Section 4]. From now on, a separable extension S/R of finite chain rings is understood to be a separable Galois extension.
The Galois group G of a separable extension S/R is isomorphic to the Galois group of the extension F q n /F q , where F q n = S/M , F q = R/m. G is cyclic with generator σ(a) = a q for a suitable primitive element a ∈ S, and {a, σ(a), . . . , σ n−1 (a)} is a free R-basis of S. Since S is also an unramified extension of R, M = Sm = Sp, and S = (1) ⊇ Sp ⊇ · · · ⊇ Sp t = (0). The automorphism groups of S are known [1,2]. Example 2. (i) The integer residue ring Z p e and the ring F p n [u]/(u e ) are finite chain rings of characteristic p, the later has nilpotency index e and residue field F p n .
(ii) A finite unital ring R is called a Galois ring if it is commutative, and its zerodivisors ∆(R) have the form pR for some prime p. (p) = Rp is the unique maximal ideal of R. Given a prime p and positive integers e, n, denote by G(p e , n) the Galois ring of characteristic p s and cardinality p en which is unique up to isomorphism. Its residue field (also called top-factor ) G(p e , n) = G(p e , n)/pG(p e , n) is the finite field F p n . 6.2. Skew-polynomials and Petit's algebras over finite chain rings. Let S be a finite chain ring with residue class field K = S/(γ) and σ ∈ Aut(S), δ a left σderivation. Consider the skew polynomial ring R = S[t; σ, δ]. Whenever S is a finite chain ring, we suppose σ((γ)) = (γ) and δ((γ)) ⊂ (γ). Then the automorphism σ induces an automorphism with σ = σ • π, and analogously δ a left σ-derivation δ : K → K. There is the canonical surjective ring homomorphism . Since S f ∼ = S af for all invertible a ∈ S, without loss of generality we consider only monic f in this section. From now on let f ∈ R = S[t; σ, δ] be monic of degree m > 1.  In particular, if S = G(p s , n) then S f has p snm elements and S f has p nm elements.
Proof. The residue class field K has q elements if |S| = q e . Since S f is a left Smodule with basis t i , 0 ≤ i ≤ m − 1, it has q em elements, analogously, S f has q m elements.
(i) S f is a unital nonassociative algebra with finitely many elements over the subring which has as kernel the two-sided ideal γS f . This induces an isomorphism of nonassociative rings:  Proof. K/F is a Galois extension with Galois group Gal(K/F ) = σ of order m.
We have f (t) = t m −d. With the assumptions in (i) resp. (ii), S f is a nonassociative cyclic division algebra over F [46] and thus the finite nonassociative ring S f is a GGR by [21,Theorem 1]. It is straightforward to see that Fix(σ) = Fix(σ) using isomorphism (1) and that S f is a lifting of its top-factor by Theorem 4.1.
Note that although the top-factor in Corollary 3 is a nonassociative cyclic algebra, it is unlikely that the algebra S f is isomorphic to a generalized nonassociative cyclic algebra as defined in Example 1 unless h = 0. If d = 1, the code is called σ-cyclic. Note that (iii) is a straightforward consequence from the fact that the k-th row of the matrix generating C is given by left multiplication of g with t k in S f , i.e. by L t k (g) = L k t (g). In particular, when δ = 0 and f (t) = t m −d, for any p ∈ S f , the matrix representing right multiplication R p with respect to the basis 1, t, . . . , t m−1 is the circulant matrix defined in [15,Definition 3.1], see also Section 5.
This is already part of [8,Theorem 2] and generalizes [13,Proposition 1]: it shows that sometimes h is a parity check polynomial for C also when f is not two-sided. Note that when we only have hg = f , h monic, and C is the code generated by g then if ch = 0 in S f , c is a codeword of C. Corollary 4. Let g = r i=0 g i t i be a monic polynomial which is a right divisor of f , such that f = gh = h g for two monic polynomials h, h ∈ S f . Let C be the cyclic (f, σ, δ)-code corresponding to g. Then the matrix representing right multiplication R h with h in S f with respect to the basis 1, t, . . . , t m−1 is a control matrix of the cyclic (f, σ, δ)-code corresponding to g.
Proof. The matrix H with ith row the vector representing For a linear code C of length m we denote by C(t) the set of skew polynomials a(t) = m−1 i=0 a i t i ∈ S f associated to the codewords (a 0 , . . . , a n ) ∈ C. For any monic f ∈ S[t; σ, δ], representing the right multiplication R g in S f by the matrix Y calculated with respect to the S-basis 1, t, . . . , t m−1 gives the injective S-linear map γ : For algebras S f which are not associative, this is not a regular representation of the algebra. However, we can prove some weaker results for special choices of f : Then the product of the m × m matrices representing R d , 0 = d ∈ S ⊂ S f , and R g for any 0 = g ∈ S f , is the matrix representing R dg , i.e. the matrix representing the right multiplication with dg in S f .
The proofs are straightforward but tedious calculations [9]. The case where f (t) = t m − d 0 ∈ S[t; σ] and S is a cyclic Galois extension of degree m over a field F with σ generating its automorphism group is already treated in [45], its proof holds analogously when S is a commutative ring with an automorphism σ of order m.
When S is a commutative unital ring, we define a map M : S f → S by for all h ∈ S f . Note that this is analogous to the definition of the reduced norm of an associative central simple algebra. We recall the following: Let A be an algebra over a ring S 0 and D a subalgebra of A, both free of finite rank as Furthermore, a map M : A → D has degree n over S 0 if M (av) = a n M (v) for all a ∈ S 0 , v ∈ A and if the map M : A × · · · × A → D defined by (1 ≤ l ≤ n) is an n-linear map over S 0 , i.e., M : A × · · · × A → D (n-copies) is an S 0 -multilinear map where M (v 1 , . . . , v n ) is invariant under all permutations of its variables.
Corollary 6. Suppose S is a commutative unital ring and both S and the algebra S f are free of finite rank as S 0 -module. For This is a direct consequence of Lemma 7.3. For properties of left semi-linear maps, especially for those of lower degree, the reader is referred to [45], [38].
Example 3. Let K/F be a cyclic Galois extension of degree m with reduced norm N K/F and reduced trace Then M : S f → S is a left semi-multiplicative map of degree m. If a ∈ K is considered as an element of S f then M (a) = N K/F (a). In particular, for m = 3 and h = h 0 + h 1 t + h 2 t 2 , we have ( [45]) Remark 6. We point out that if S = (K/F, , c) is a suitable cyclic division algebra with norm N S/F , we can describe the right multiplication with h by an mn × mn matrix X(h) with entries in K as described in Remark 4 (ii), and define a map which is also left-semilinear for suitable f (t) = t m − d (cf. [37,Remark 19] where we look at the matrix representing left multiplication instead, since we are dealing with the opposite algebra there). Again the map M can be seen as a generalization of the norm of an associative central simple algebra and for all x ∈ S for suitably chosen S 0 -algebras S f , for details see [37]. 7.2. Codes over finite chain rings. Let S be a finite chain ring and σ an automorphism of S. The S[t; σ]-module S[t; σ]/S[t; σ]f is increasingly favored for linear code constructions over S, with f a monic polynomial of degree m (usually f (t) = t m − d), cf. for instance [4], [6], [26]. For code constructions, we generally look at reducible skew polynomials f .
We take the setup discussed in [4], [6], [26], where the S[t; σ]-module S[t; σ]/ S[t; σ]f is employed for linear code constructions, and discuss on some examples how the results mentioned previously fit into our view of equipping S[t; σ]/S[t; σ]f with a nonassociative algebra structure: • In [26, Theorem 2.2], it is shown that a code of length n is σ-constacyclic if and only if the skew polynomial representation associated to it is a left ideal in S f , again assuming S f to be associative, i.e. f (t) = t m − d ∈ S[t; σ] with d ∈ S invertible, to be two-sided, and S to be a finite chain ring.
• In [6, Proposition 2.1], it is shown that any right divisor g(t) of f (t) = t m −d ∈ S[t; σ] generates a principal left ideal in S f , provided that f is a monic twosided element and assuming S is a Galois ring. The codewords associated with the elements in the ideal Rg form a code of length m and dimension m − degg. This also holds in the nonassociative setting, so we can drop the assumption in [6, Proposition 2.1] that f needs to be a monic central element, see Corollary 5. • In [4, Theorem 2] (or similarly in [26, 3.1]), it is shown that if a skew-linear code C is associated with a principal left ideal, then C is an S-free module if and only if g is a right divisor of f (t) = t m − 1, again assuming S to be Galois, and f two-sided. This is generalized in Proposition 4, resp. Corollary 5. • For f (t) = t m − d ∈ F q [t; σ], the (σ, d)-circulant matrix M σ d in [15] is the matrix representing R g in the algebra S f calculated with respect to the basis 1, t, . . . , t m−1 . Therefore [15,Theorem 3.6] states that for associative algebras S f , right multiplication gives the right regular representation of the algebra, so that the product of the matrix representing R h , and the one representing R g , for any 0 = h ∈ S f , 0 = g ∈ S f , is the matrix representing R hg in S f . The fact that γ is injective and additive is observed in [15,  Moreover, the matrix equation in [15, Theorem 5.6 (1)] can be read as follows: if t n − a = hg and c = γ(a, g), then the matrix representing the right multiplication with the element g(t) ∈ R n in the algebra S f where f (t) = t n − a ∈ F q [t; σ], equals the transpose of the matrix representing the right multiplication with an element g (t) ∈ S f1 where f 1 (t) = t n − c −1 ∈ F q [t; σ]. This suggests an isomorphism between S f1 = F q [t; σ]/F q [t; σ]f 1 and the opposite algebra of S f = F q [t; σ]/F q [t; σ]f .

Conclusion and further work
This paper proposes a more general way of looking at cyclic (f, σ, δ)-codes using nonassociative algebras, and unifies different ways of designing cyclic linear (f, σ, δ)codes in a general, nonassociative theory. Connections between the algebras and some fast-decodable space-time block code designs are pointed out along the way.
It is well known that for any f ∈ R = S[t; σ, δ], R/Rf is an R-module with the module structure given by the multiplication g(h + Rf ) = gh + Rf = r + Rf if r is the reminder of gh after right dividing by f . This is exactly the multiplication which makes the additive group {g ∈ R | deg(g) < m} into a nonassociative algebra when f has an invertible leading coefficient. Thus one might argue that the introduction of the nonassociative point of view we suggested here seems to make things only more complicated that actually needed and not necessarily better.
The full benefits of this approach for coding theory might only become visible once more work has been done in this direction. Using the nonassociative Petit algebras S f over number fields allows us for instance to show how certain cyclic (f, σ, δ)-codes over finite rings canonically induce a Z-lattice in R N . The observations in [13, Section 5.2, 5.3] hold analogously for our nonassociative algebras and explain the potential of the algebras S f for coset coding in space-time block coding, in particular for wiretap coding, cf. [35]. Previous results for lattices obtained from σ-constacyclic codes related to associative cyclic algebras by Ducoat and Oggier [13] are obtained as special cases.