2.1. Nonassociative algebras

Let $F$ be a field. We call $A$ an algebra over $F$ if there exists an $F$ -bilinear map $A\times A\to A$ , $(x,y) \mapsto x \cdot y$ , 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 $x\in A$ . We will only consider unital algebras. A nonassociative algebra $A\not =0$ is called a division algebra if for any $a\in A$ , $a\not =0$ , the left multiplication with $a$ , $L_a(x)=ax$ , and the right multiplication with $a$ , $R_a(x)=xa$ , are bijective. If $A$ is finite-dimensional as an $F$ -vector space, then $A$ is a division algebra if and only if $A$ has no zero divisors. The left nucleus of $A$ is defined as ${\rm Nuc}_l(A) = \{ x \in A \, \vert \, [x, A, A] = 0 \}$ , the middle nucleus of $A$ is $\textrm{Nuc}_m(A) = \{ x \in A \, \vert \, [A, x, A] = 0 \}$ , and the right nucleus of $A$ is $\textrm{Nuc}_r(A) = \{ x \in A \, \vert \, [A,A, x] = 0 \}$ , where $[x, y, z] = (xy) z - x (yz)$ is the associator. $\textrm{Nuc}_l(A)$ , $\textrm{Nuc}_m(A)$ , and $\textrm{Nuc}_r(A)$ are associative subalgebras of $A$ . Their intersection $\textrm{Nuc}(A) = \{ x \in A \, \vert \, [x, A, A] = [A, x, A] = [A,A, x] = 0 \}$ is the nucleus of $A$ . $\textrm{Nuc}(A)$ is an associative subalgebra of $A$ , and $x(yz) = (xy) z$ whenever one of the elements $x, y, z$ is in $\textrm{Nuc}(A)$ . The center of $A$ is $\textrm{C}(A)=\{x\in \text{Nuc}(A)\,|\, xy=yx \text{ for all }y\in A\}$ .

Let $A$ be a finite-dimensional central simple associative algebra over $F$ of degree $d$ and let $\overline{F}$ denote the algebraic closure of $F$ . Then, $A\otimes _F \overline{F} \cong M_d(\overline{F})$ , so that we can fix an embedding $A \longrightarrow M_d(\overline{F})$ and view every $a\in A$ as a matrix in $M_d(\overline{F})$ . The characteristic polynomial

\begin{equation}m_a(X)=X^d-s_1(a)X^{d-1}+s_2(a)X^{d-2}-\cdots+(\!-\!1)^ds_d(a),\end{equation}

of $a\in A$ has coefficients in $F$ and is independent of the choice of the embedding. The coefficient $N_A(a)=s_d(a)$ is called the reduced norm of $a$ [Reference Knus, Merkurjev, Rost and Tignol20]. Let $K/F$ be a cyclic Galois extension of degree $d$ with Galois group $\textrm{Gal}(K/F)=\langle \gamma \rangle$ and norm $N_{K/F}$ . Let $c\in F^\times$ . An associative cyclic algebra $(K/F,\gamma,c)$ of degree $d$ over $F$ is a $d$ -dimensional $K$ -vector space

\begin{equation*} (K/F,\gamma,c)=K \oplus eK \oplus e^2 K\oplus \dots \oplus e^{d-1}K, \end{equation*}

with multiplication given by the relations $ e^d=c,\,le=e\sigma (l),$ for all $l\in K$ . $(K/F,\gamma,c)$ is a division algebra for all $c\in F^\times$ , such that $c^s\not \in N_{K/F}(K^\times )$ for all $s$ which are prime divisors of $d$ , $1\leq s\leq d-1$ .

2.2. MRD codes

Let $K$ be a field. A code is a set of matrices $\mathcal{C}\subset M_{n, m}(K)$ . Let $L\subset K$ be a subfield, then $\mathcal{C}$ is $L$ -linear if $\mathcal{C}$ is a vector space over $L$ . A rank metric code is a code $\mathcal{C}\subset M_{n, m}(K)$ equipped with the rank distance function $d(X,Y)=\textrm{rank}(X-Y).$ Define the minimum distance of a rank metric code $\mathcal{C}$ as

\begin{equation*}d_{\mathcal {C}}=\text {min}\lbrace d(X,Y)\mid X,Y\in \mathcal {C}, X\neq Y\rbrace .\end{equation*}

An $L$ -linear rank metric code $\mathcal{C}$ satisfies the Singleton-like bound

\begin{equation*}\text { dim}_{L}(\mathcal {C})\leq n(m-d_{\mathcal {C}}+1)[K:L],\end{equation*}

where $\text{dim}_L(\mathcal{C})$ is the dimension of the $L$ -vector space $\mathcal{C}$ [Reference Augot, Loidreau and Robert2, Proposition 6].

An $L$ -linear rank metric code attaining the Singleton-like bound is called a maximum rank distance code or MRD code (for MRD codes over cyclic field extensions see [Reference Augot, Loidreau and Robert2]).

If now $B$ is a not necessarily commutative division algebra then more generally, we again define a code as a set of matrices $\mathcal{C}\subset M_{n, m}(B)$ . Let $B'\subset B$ be a subalgebra, then $\mathcal{C}$ is $B'$ -linear (or simply linear), if $\mathcal{C}$ is a right $B'$ -module.

A rank metric code $\mathcal{C}\subset M_{n, m}(B)$ is a code together with the distance function

\begin{equation*}d(X,Y)=\text {colrank}(X-Y),\end{equation*}

for all $X,Y\in M_{n,m}(B)$ , where $\textrm{colrank}$ is the column rank of $A$ (the rank of the right $B$ -module generated by the columns of $A$ ). A matrix in $M_{n,m}(B)$ has column rank at most $m$ ; any matrix which attains this bound is said to have attained full column rank. The minimum distance of a rank metric code $\mathcal{C}\subset M_{n,m}(B)$ is defined as

\begin{equation*}d_{\mathcal {C}}=\text {min}\lbrace d(X,Y)\mid X,Y\in \mathcal {C}, X\neq Y\rbrace .\end{equation*}

To our knowledge, such codes $\mathcal{C}\subset M_{n, m}(B)$ have not previously been considered in the literature.

2.3. Skew polynomial rings

In the following, let $D$ be a central simple division algebra of degree $d$ over its center $C$ , $\sigma$ a ring endomorphism of $D$ and $\delta\;:\;D\rightarrow D$ a left $\sigma$ -derivation, i.e. an additive map such that $\delta (ab)=\sigma (a)\delta (b)+\delta (a)b$ for all $a,b\in D$ . The skew polynomial ring $D[t;\;\sigma,\delta ]$ is the set of skew polynomials $g(t)=a_0+a_1t+\dots +a_nt^n$ with $a_i\in D$ , with term-wise addition and multiplication defined via $ta=\sigma (a)t+\delta (a)$ for all $a\in D$ [Reference Ore22]. Define $\textrm{Fix}(\sigma )=\{a\in D\,|\, \sigma (a)=a\}$ and $\textrm{Const}(\delta )=\{a\in D\,|\, \delta (a)=0\}$ . If $\delta =0$ , define $D[t;\;\sigma ]=D[t;\;\sigma,0]$ . If $\sigma =id$ , define $D[t;\;\delta ]=D[t;\;id,\delta ]$ .

For $f(t)=a_0+a_1t+\dots +a_nt^n\in R=D[t;\;\sigma,\delta ]$ with $a_n\not =0$ , we define the degree of $f$ as $ \textrm{deg}(f)=n$ and $\textrm{deg}(0)=-\infty$ . A skew polynomial $f\in R$ is irreducible if it is not a unit and it has no proper factors, i.e if there do not exist $g,h\in R$ with $1\leq \textrm{deg}(g), \textrm{deg }(h)\lt \textrm{deg}(f)$ such that $f=gh$ [Reference Jacobson18, p. 2 ff.]. We call $f\in R$ right-invariant if $Rf$ is a left and a right ideal in $R$ , and a two-sided maximal element, if $f$ is right-invariant and $Rf$ is a nonzero maximal ideal in $R$ (equivalently, if $f\not =0$ and $R/Rf$ is a simple ring) [Reference Jacobson18, p. 13]. Two nonzero skew polynomials $f_1,f_2 \in R$ are similar, written $f_1\sim f_2$ , if $R/Rf_1 \cong R/Rf_2$ [Reference Jacobson18, p. 11].

A skew polynomial $f \in R$ is bounded if there exists a nonzero polynomial $f^* \in R$ such that $Rf^*$ is the largest two-sided ideal of $R$ contained in $Rf$ . The polynomial $f^*$ is uniquely determined by $f$ up to scalar multiplication by elements of $D^\times$ and is called a bound of $f$ .

If $f\in R$ has degree $m$ , then for all $g\in R$ of degree $l\geq m$ , there exist uniquely determined $r,q\in R$ with $\textrm{deg}(r)\lt \textrm{deg}(f)$ , such that $g=qf+r.$ Let $\textrm{mod}_r f$ denote the remainder of right division by $f$ . The skew polynomials $R_m=\{g\in R\,|\, \textrm{deg}(g)\lt m\}$ of degree less that $m$ canonically represent the elements of the left $R$ -modules $R/Rf$ . Furthermore, $R_m$ together with the multiplication $g\circ h=gh \, \textrm{mod}_r f$ is a unital nonassociative algebra $S_f=(R_m,\circ )$ over $F_0=\{a\in D\,|\, ah=ha \text{ for all } h\in S_f\}=\textrm{Comm}(S_f)\cap D$ , called a Petit algebra. When the context is clear, we simply use juxtaposition for multiplication in $S_f$ . Note that $ C(D)\cap \textrm{Fix}(\sigma )\cap \textrm{Const}(\delta )\subset F_0.$ For all $a\in D^\times$ , we have $S_f = S_{af}$ ; thus, without loss of generality we can assume $f$ is monic when working with Petit algebras $S_f$ . If $f$ has degree 1 then $S_f\cong D$ .

Proof. Suppose $\gamma$ is a right divisor of $g$ . Then, $g=\delta \gamma$ for some $\delta \in R$ . As $g$ lies in the center of $R$ , we have $\delta g = g\delta =\delta \gamma \delta$ . This rearranges to $0=\delta g-\delta \gamma \delta =\delta (g-\gamma \delta ).$ As $R$ contains no zero divisors and $\delta \neq 0$ , it follows that $g=\gamma \delta$ .

2.4. The minimal central left multiple of f $\boldsymbol\in$ D[t; $\boldsymbol\sigma$ ]

From now on let, $\sigma$ be an automorphism of $D$ of finite order $n$ modulo inner automorphisms, i.e. $\sigma ^n=i_u$ for some inner automorphism $i_u(z)=uzu^{-1}$ . Then, the order of $\sigma |_C$ is $n$ . W.l.o.g., we choose $u\in \textrm{Fix}(\sigma )$ . Let $R=D[t;\;\sigma ]$ and define $F=C\cap \textrm{Fix}(\sigma )$ . $R$ has center

\begin{equation*}C(R) = F[u^{-1}t^n]=\left\{\sum _{i=0}^{k}a_i(u^{-1}t^n)^i\,|\, a_i\in F \right\}\cong F[x]\end{equation*}

with $x=u^{-1}t^n$ [Reference Jacobson18, Theorem 1.1.22]. All polynomials $f\in R$ are bounded.

For any $f \in R=D[t;\;\sigma ]$ with a bound in $C(R)$ , we define the minimal central left multiple $mclm(f)$ of $f$ in $R$ to be the unique polynomial of minimal degree $h \in C(R)=F[u^{-1}t^n]$ such that $h = gf$ for some $g \in R$ , and such that $h(t)=\hat{h}(u^{-1}t^n)$ for some monic $\hat{h}(x) \in F[x]$ . Define $E_{\hat{h}}=F[x]/(\hat{h}(x))$ . If $f$ has nonzero constant term, then $f^*\in C(R)$ [Reference Gòmez-Torrecillas, Lobillo and Navarro11, Lemma 2.11]). From now on, we assume that $f$ has nonzero constant term and denote by $h \in C(R)$ , $h(t)=\hat{h}(u^{-1}t^n)$ , the minimal central left multiple of $f$ . Then, $h$ equals the bound of $f$ up to a scalar multiple from $D$ . If $f$ is irreducible in $R$ , then $\hat{h}(x)$ is irreducible in $F[x]$ . If $\hat{h}\in F[x]$ is irreducible, then $f=f_1\cdots f_r$ for irreducible $f_i\in R$ such that $f_i\sim f_j$ for all $i,j$ ([Reference Owen23], cf. [Reference Thompson and Pumplün36]).

This follows easily from [Reference Carcanague7, p. 9, Corollary 2] and [Reference Jacobson18, Theorem 1.2.9].

The quotient algebra $ R/Rh$ has center $ C(R/Rh)\cong F[x]/ (\hat{h}(x) )$ , cf. [Reference Gòmez-Torrecillas, Lobillo and Navarro11, Lemma 4.2]. Define $E_{\hat{h}}=F[x]/ (\hat{h}(x) )$ . Suppose that $\hat{h}(x) \neq x$ and that $\hat{h}$ is irreducible in $F[x]$ . Then, $h$ generates a maximal two-sided ideal $Rh$ in $R$ [Reference Jacobson18, p. 16] and $R/Rh$ is simple over its center $ E_{\hat{h}}$ .

Theorem 3 [Reference Owen23]. Let $f \in R=D[t;\;\sigma ]$ be monic and irreducible of degree $m\gt 1$ with minimal central left multiple $h(t)=\hat{h}(u^{-1}t^n)$ . Then,, $\textrm{Nuc}_r(S_f)$ is a central division algebra over $E_{\hat{h}}$ of degree $s=dn/k$ , where $k$ is the number of irreducible factors of $h$ in $R$ , and

\begin{equation*} R/Rh \cong M_k(\textrm {Nuc}_r(S_f)).\end{equation*}

In particular, this means $\textrm{deg}(\hat{h})=\frac{dm}{s}$ , $\textrm{deg}(h)=km=\frac{dn m}{s}$ , and

\begin{equation*}[\textrm {Nuc}_r(S_f)\;:\;F]= s^2\cdot \frac {dm}{s}=dms.\end{equation*}

Moreover, $s$ divides $gcd(dm,dn)$ . If $f$ is not right-invariant, then $k\gt 1$ and $s\not =dn$ .

We know that $[S_f\;:\;F]=[S_f\;:\;C][C\;:\;F]=d^2m\cdot n$ . Since $\textrm{Nuc}_r(S_f)$ is a subalgebra of $S_f$ , comparing dimensions we obtain that

\begin{equation*}d^2mn=[S_f\;:\;F]=[S_f\;:\;\textrm {Nuc}_r(S_f)]\cdot [\textrm {Nuc}_r(S_f)\;:\;F]=k\cdot dms,\end{equation*}

that is $[S_f\;:\;\textrm{Nuc}_r(S_f)]=k.$

If $f$ is not right-invariant which is equivalent to $S_f$ being not associative, which in turn is equivalent to $k\gt 1$ , then $s\not =dn$ looking at the degree of $h$ . Note that $\textrm{deg}(h)=dn m$ is the largest possible degree of $h$ .

All of the above applies in particular to the special case that $D$ is a finite field extension $K$ of $C$ of degree $n$ , and $\sigma \in \textrm{Aut}(K)$ has order $n$ . Then, $R=K[t;\;\sigma ]$ has center $C(R) = F[t^n]=\{\sum _{i=0}^{k}a_i(t^n)^i\,|\, a_i\in F \}= F[x]$ where $F= \textrm{Fix}(\sigma )$ [Reference Jacobson18, Theorem 1.1.22].

5.1. The algebra (D(x), $\widetilde{\sigma }$ , ux)

Let $C/F$ be a finite cyclic field extension of degree $n$ with $\textrm{Gal}(C/F) = \langle \sigma \rangle$ . Let $D$ be a finite-dimensional division algebra of degree $d$ with center $C$ and suppose that $\sigma$ extends to a $C$ -algebra automorphism of $D$ that we call $\sigma$ , too. Let $R=D[t;\;\sigma ]$ as in Section 3. Then, there exists $u \in D^\times$ such that $\sigma ^n = i_u$ and $\sigma (u) = u$ . These two relations determine $u$ up to multiplication with elements from $F^\times$ [Reference Pierce25, Lemma 19.7].

The quotient algebra $(D,\sigma,a)=D[t;\;\sigma ]/(t^n-a)D[t;\;\sigma ]$ , where $f(t)=t^n-a\in D[t;\;\sigma ]$ with $d\in F^\times$ , is called a generalized cyclic algebra. The special case where $D=C$ yields the cyclic algebra $(C/F,\gamma,a)$ [Reference Jacobson18, p. 19].

Let $D(t;\;\sigma ) = \{f/g \,|\, f \in D[t;\;\sigma ], g \in C(D[t;\;\sigma ])\}$ be the ring of central quotients of $D[t;\;\sigma ]$ . Let $\widetilde{\sigma }$ denote the extension of $\sigma$ to $D(x)$ that fixes $x$ [Reference Hanke14, Lemma 2.1]. Then, $C(D(t;\; \sigma )) = \textrm{Quot}(C(D[t;\;\sigma ])) = F(x)$ , $x =u^{-1}t^n$ , is the center of $D(t;\;\sigma )$ , where $\textrm{Quot}(U)$ denotes the quotient field of an integral domain $U$ . More precisely, $D(t;\;\sigma ) \cong (D(x), \widetilde{\sigma }, ux )$ is a generalized cyclic algebra of degree $dn$ over its center $F(x)$ and a division algebra [Reference Hanke14, Theorems 2.2, 2.3].

Let $N$ be the reduced norm of $(D(x), \widetilde{\sigma }, ux )$ .

Proof. If $f=gp$ for $g,p\in R$ then $N(f)=N(g)N(p)$ is reducible in $F[x],$ since both $N(g)$ and $N(p)$ lie in $F[x]$ , which immediately yields the assertion.

From now on, we assume that

\begin{equation*}D=(E/C,\gamma,a) \text { is a cyclic division algebra over } C \text { of degree } d,\end{equation*}

\begin{equation*}\sigma |_E\in \textrm {Aut}(E) \text { such that }\gamma \circ \sigma =\sigma \circ \gamma \text { and } u\in E.\end{equation*}

Then, $\sigma |_E$ has order $n$ . Write $m=kn+r$ for some $0\leq r\lt n$ . Let $f=\sum _{i=0}^ma_it^i\in R$ be a polynomial such that $a_0\not =0$ and $h\in R$ be the minimal central left multiple of $f$ in $R$ .

Theorem 9 [Reference Thompson and Pumplün36]. For $f\in E[t;\;\sigma ]\subset D[t;\;\sigma ]$ , we have

\begin{equation*}N(f(t))=N_{E/F}(a_0)+\dots +(\!-\!1)^{dr(n-1)}N_{E/F}(a_m)N_{E/C}(u)^{m}x^{dm}.\end{equation*}

Theorem 10. Suppose that $\textrm{deg}(h)=dmn$ .

1. (i) [Reference Thompson and Pumplün36, Theorem 14 (i)] If $\hat{h}$ is irreducible in $F[x]$ , then $f$ is irreducible in $R$ .

2. (ii) [Reference Thompson and Pumplün36, Theorem 14 (ii)] If $f$ is irreducible, then $N(f)$ is irreducible in $F[x]$ .

3. (iii) If $f\in E[t;\;\sigma ]$ , then $N(f)= (\!-\!1)^{dr(n-1)}N_{E/F}(a_m)N_{E/C}(u)^{m} \hat{h}$ and

\begin{equation*}N_{E/F}(a_0)=(\!-\!1)^{dr(n-1)}N_{E/F}(a_m)N_{E/C}(u)^{m}h_0,\end{equation*}

if $h_0$ denotes the constant term of $\hat{h}$ .

Proof. (iii) By Theorem 9, we have $\textrm{deg}(N(f))=dmn$ in $R$ . $N(f)$ is a two-sided multiple of $f$ in $R$ ; therefore, the bound $f^*$ of $f$ divides $N(f)$ in $R$ . Since $(f,t)_r=1,$ $f^*\in C(R)$ and therefore $f^*$ equals $h$ up to some factor in $F^\times$ . Thus, $h(t)=\hat{h}(u^{-1}t^n)$ must divide $N(f)$ in $R$ . Write $N(f)=g (t)h(t)$ for some $g\in R$ . Comparing degrees in $R$ , we obtain $\textrm{deg} N(f)=\textrm{deg}(g(t))+dmn=dmn$ , which implies $\textrm{deg}(g)=0$ , i.e. $g(t)=a\in A^\times$ . This implies that $N(f)=ah(t)=a \hat{h}(u^{-1}t^n)$ . Comparing highest coefficients of $N(f)$ and $a\hat{h}$ yields that $a=(\!-\!1)^{dr(n-1)}N_{E/F}(a_m)N_{E/C}(u)^{m}$ by Theorem 9, so that comparing constant terms we get that $N_{E/F}(a_0)=(\!-\!1)^{dr(n-1)}$ $N_{E/F}(a_m)N_{E/C}(u)^{m}h_0$ , if $h_0$ is the constant term of $\hat{h}(x)$ .

Theorem 11. Let $f\in E[t;\;\sigma ]\subset R$ be monic and irreducible of degree $m$ . Let $\textrm{deg}(\hat{h})=dm$ and suppose that all the monic polynomials similar to $f$ lie in $E[t;\;\sigma ]$ . If $g$ is a monic divisor of $h$ in $R$ of degree $lm$ , then

\begin{equation*}N_{E/F}(g_0)=N_{E/F}(a_0)^l.\end{equation*}

Proof. We know that $h(t)=\hat{h}(u^{-1}t^n)$ , with $\hat{h}(x)$ irreducible in $F[x]$ , since $f$ is irreducible. Thus, $h$ is a t.s.m. element in Jacobson’s terminology [Reference Jacobson18] and the irreducible factors $f_1(t),\dots, f_k(t)$ of any decomposition of $h(t)$ are all similar and are all similar to $f$ , as $f$ must be one of them by the definition of $h$ . Now, $g(t)$ is a monic divisor of $h$ . Thus, we can decompose $g(t)$ into a product of irreducible factors and up to similarity the irreducible factors of $g$ will be the same as suitably chosen irreducible factors of $h$ by [Reference Jacobson18, Theorem 1.2.9.]. Hence, w.l.o.g. $g=f_1 f_2\cdots f_l$ , where the $f_i$ are irreducible in $R$ and $f_i$ is similar to $f$ for all $i=1,2,\dots,l$ [Reference Jacobson18, Theorem 1.2.19]. Thus by Lemma 2, the minimal central left multiple of each $f_i$ is equal to $h$ . Since $f$ is monic, we may assume w.l.o.g. that all $f_i$ are monic. By Theorem 10 and since all $f_i\in E[t;\;\sigma ]$ by our assumption, this implies that $N_{E/F}(f_i(0))=(\!-\!1)^{dm(n-1)}N_{E/C}(u)^{m}h_0=N_{E/F}(a_0)$ . As the constant term of $g$ is equal to $\prod _{i=1}^l f_i(0)$ , we see that

\begin{align*}N_{E/F}(g_0)&=\prod _{i=1}^l N_{E/F}(f_i(0))=[ (\!-\!1)^{dm(n-1)}N_{E/C}(u)^{m}h_0]^l\\[2pt] &=(\!-\!1)^{l dm(n-1)}N_{E/C}(u)^{lm}h_0^l=N_{E/F}(a_0)^l.\end{align*}

We are not able to say if the assumptions on the $f_i$ ’s in the above result are empty or trivial.

5.2. The algebra $(\boldsymbol{{K}}(\boldsymbol{{x}})/\boldsymbol{{F}}(\boldsymbol{{x}}),\widetilde{\boldsymbol\sigma },\boldsymbol{{x}})$

Let $K/F$ be a cyclic field extension of degree $n$ with $\textrm{Gal}(K/F)=\langle \sigma \rangle$ , $R=K[t;\;\sigma ]$ and $x=t^n$ . We now look at the cyclic algebra $(K(x)/F(x),\widetilde{\sigma },x)$ (this case corresponds to $D=C$ in the previous Section). Let $N$ be the reduced norm of $(K(x)/F(x),\widetilde{\sigma },x)$ over $F(x)$ (cf. also [Reference Jacobson18, Proposition 1.4.6]). We have $\widetilde{\sigma }|_K=\sigma$ , and $N$ is a nondegenerate form of degree $n$ . Let $f=\sum _{i=0}^ma_it^i\in R$ be a polynomial of degree $m$ such that $a_0\not =0$ and $h\in R$ be the minimal central left multiple of $f$ in $R$ . Then $N(f(t))=N_{K/F}(a_0)+\dots + (\!-\!1)^{m(n-1)}N_{K/F}(a_m)x^m$ [Reference Thompson and Pumplün36, Theorem 3].

Theorem 12 (iii) is proved analogously as Theorem 10 (iii).

Theorem 13 (cf. [Reference Sheekey34, Theorem 5] for finite fields, the proof is the same). Suppose that $f$ is not right-invariant. If $\textrm{deg}(h)=mn$ and $g$ is a monic divisor of $h(t)$ in $R$ of degree $ml$ , then

\begin{equation*}N_{K/F}(g_0)=N_{K/F}(a_0)^l.\end{equation*}

6.1. The case that f $\in$ D[t; $\sigma$ ]

Let $f\in R=D[t;\;\sigma ]$ be a monic polynomial of degree $m$ . Let $\rho \in \textrm{Aut }(D)$ , $\nu \in D$ and $F'=\textrm{Fix}(\rho )\cap F$ where $F=C\cap \textrm{Fix}(\sigma )$ . Let $b(t), c(t)\in R_m=\{g\in R\,|\, \textrm{deg}(g)\lt m\}$ and $b_0$ be the constant term of $b(t)$ . Then, the multiplication defined via

\begin{equation}b(t)\circ c(t)=(b(t)+\nu\rho(b_0)t^m)c(t)\hspace{0.167em}\hspace{0.167em}{\text{mod}}_rf,\end{equation}

makes $R_m$ into a non-unital nonassociative ring $(R_m,\circ )$ . When the context is clear, we will drop the $\circ$ notation and simply use juxtaposition. $(R_m,\circ )$ is an algebra over $F'$ .

Example 14. If $f(t)=t-c\in D[t;\;\sigma ]$ for some $c\in D$ , $\nu \neq 0$ , then $(R_m,\circ )$ has the multiplication

\begin{align*} a\circ b&=(a+\nu\rho(a)t)b)\hspace{0.222em}\hspace{0.167em}\hspace{0.167em}{\text{mod}}_rf\\[5pt] &=ab+\nu\rho(a)\sigma(b)t\hspace{0.222em}\hspace{0.167em}\hspace{0.167em}{\text{mod}}_rf\\[5pt] &=ab+\nu\rho(a)\sigma(b)c,\end{align*}

for all $a,b\in D$ . This generalizes the algebras studied in [Reference Pumplün30]. If $R=K[t;\;\sigma ]$ for some finite field extension $K/F$ ; this is the multiplication of Albert’s twisted semifields [Reference Albert1]. If $F/F'$ is finite and $(R_m,\circ )$ is not a division algebra, $a\circ b=0$ for some nonzero $a,b\in D$ , if and only if $ ab=-\nu \rho (a)\sigma (b)c$ . Taking norms of both sides and canceling $N_{D/F'}(ab)$ on both sides, we obtain that $N_{D/F'}(\!-\!\nu c)=(\!-\!1)^{d^2n[F\;:\;F']} N_{D/F'}(\nu c)=1.$ Thus, if $F/F'$ is finite and $N_{D/F'}(\nu c)\neq (\!-\!1)^{d^2n[F\;:\;F']}$ then $(R_m,\circ )$ is a division algebra.

From now on for the rest of the paper, we again assume that $f$ is an irreducible monic polynomial of degree $m\gt 1$ , $(f,t)_r= 1$ , and that $h$ is the minimal central left multiple of $f$ . Let $F/F'$ be finite-dimensional, and

\begin{equation*}P=\lbrace d_0+d_1t+\dots +d_{lm-1}t^{lm-1}+\nu \rho (d_0)t^{lm}\,|\, d_i\in D\rbrace \subset D[t;\;\sigma ].\end{equation*}

Note that $f$ may be right-invariant.

Proof. Suppose that there are $b(t)=b_0+b_1t+\dots +b_{m-1}t^{m-1}, c(t)\in R_m$ , such that

\begin{equation*}b(t)\circ c(t)=(b(t)+\nu \rho (b_0)t^m) c(t)\textrm {mod}_r f =0.\end{equation*}

Then, there exists $g\in R_m$ such that $(b(t)+\nu \rho (b_0)t^m) c(t)=g(t)f(t).$ Since $f$ is irreducible and of degree $m$ , while $deg(c) \lt m$ , $f$ must be similar to an irreducible factor of $b(t)+\nu \rho (b_0)t^m$ , because of the uniqueness of an irreducible decomposition in $R$ up to similarity. But $b(t)+\nu \rho (b_0)t^m$ has degree at most $m$ , so $f$ is similar to $b(t)+\nu \rho (b_0)t^m$ . Thus, $b(t)+\nu \rho (b_0)t^m$ must have degree $m$ and be irreducible as well. Hence if $b(t)+\nu \rho (b_0)t^m$ is not similar to $f$ then $b(t)+\nu \rho (b_0)t^m$ is not a left zero divisor in $(R_m, \circ )$ . This happens for instance, if $\nu =0$ or if $b(t)+\nu \rho (b_0)t^m$ is reducible. Moreover, $(R_m, \circ )$ is a division algebra if $P$ does not contain any polynomial similar to $f$ .

We are again not able to say if the assumptions on the $f_i$ ’s in the following result are empty or trivial.

Theorem 16. Let $D=(E/C,\gamma,a)$ be a cyclic division algebra over $C$ of degree $d$ such that $\sigma |_E\in \textrm{Aut}(E)$ and $\gamma \circ \sigma |_E=\sigma |_E\circ \gamma$ . Suppose that $\sigma ^n(z)=u^{-1}zu$ with $u\in E$ .

Let $f(t)=\sum _{i=0}^ma_it^i\in E[t;\;\sigma ]\subset D[t;\;\sigma ]$ be monic and irreducible, and let $\textrm{deg}(h)=dmn$ . Suppose that all monic $f_i$ similar to $f$ lie in $E[t;\;\sigma ]$ . Then, $(R_m, \circ )$ is a division algebra over $F'$ , if one of the following holds: (i) $\nu \not \in E$ and $\rho |_E\in \textrm{Aut}(E)$ . (ii) $\nu \in E^\times$ and $\rho |_E\in \textrm{Aut}(E)$ , such that

\begin{equation*}N_{E/F'}(a_0) N_{E/F'}(\nu )\neq 1.\end{equation*}

Note that our global assumption that $\sigma ^n(z)=u^{-1}zu$ for all $z\in D$ , so that $\sigma ^n(e)=u^{-1}eu=e$ for all $e\in E$ , forces $(\sigma |_E)^n=id$ .

Proof. By Theorem 15, $(R_m, \circ )$ is a division algebra, if the set $P$ with $l=1$ does not contain any polynomial similar to $f$ . All polynomials similar to $f$ are irreducible factors of $h(t)$ , so $(R_m, \circ )$ is a division algebra, if $P$ does not contain any irreducible factor of $h(t)$ . Suppose that $P$ contains an irreducible factor $g$ of $h$ with constant term $g_0$ . Then, $g$ has degree $m$ as it is similar to $f$ . Let $g_{m} t^{m}$ be its highest coefficient, so that $g_{m}^{-1}g$ is a monic divisor of $h$ .

By Theorem 10 and since $g\in E[t;\;\sigma ]$ by assumption, this implies

\begin{equation}N_{E/F}(g_0g_m^{-1})=(\!-\!1)^{m(n-1)}h_0=N_{E/F}(a_0),\end{equation}

and in particular, that $g_0$ and $g_m$ are both nonzero. Since $g\in P$ , we also have $g_{m}=\nu \rho (g_0)$ . Suppose $\nu \not \in E$ and $\rho (E)\subset E$ . Since the coefficients of the $f_i$ all lie in $E$ , we have $g_{m}\not =\nu \rho (g_0)$ which yields a contradiction. Hence, there is no divisor $g$ of $h$ in $P$ and $S$ is a division algebra. Suppose that $\nu \in E^\times$ and $\rho (E)\subset E$ . Substituting $g_{m}=\nu \rho (g_0)$ into the above equation yields

\begin{equation*}N_{E/F}(g_0)=N_{E/F}(a_0) N_{E/F}(\nu \rho (g_0)).\end{equation*}

Applying $N_{F/F'}$ to both sides implies that

\begin{equation*}N_{E/F'}(g_0)=N_{F/F'}(N_{E/F}(a_0)) N_{E/F'}(\nu \rho (g_0)).\end{equation*}

Now $N_{E/F'}(\rho (g_0))=N_{E/F'}(g_0)$ , so we can cancel the nonzero term $N_{E/F'}(g_0)$ to obtain $1=N_{E/F'}(a_0)$ $N_{E/F'}(\nu ).$

Remark 17. Let $S=S_{n,m,1}(\nu,\rho, f)=\lbrace a+Rh\,|\, a\in P\rbrace$ . We can use $\mathcal{C}(S)\subset M_k(B)$ to define a multiplication on $B^m$ . As $\textrm{dim}_F(D)=d^2n$ and $\textrm{dim}_F(B)=d^2mn/k$ , there exists an $F$ -vector space isomorphism between $D^m$ and $B^k$ . Similarly, there exists an isomorphism $G\;:\;V_f\to B^k$ , $G(a+Rf)=\underline{a}$ . Define $\ast\;:\;B^k\times B^k\to B^k$ by

\begin{equation*}\underline {a}\ast \underline {b}=M_a\cdot \underline {b}\end{equation*}

for all $\underline{a},\underline{b}\in B^k$ , where $M_a\in \mathcal{C}(S)$ is the representation of the map $L_{a(t)+\nu \rho (a_0)t^m}\in \textrm{End}_B(R/Rf)$ induced by $G$ . (Each $a\in R_m$ corresponds to a map $L_{a(t)+\nu \rho (a_0)t^m}$ . As $\textrm{End}_B(R/Rf)\cong M_k(B)$ and $\textrm{dim}(R_m)=\textrm{dim}(\mathcal{C}(S))$ , there is a canonical bijection between $L_{a(t)+\nu \rho (a_0)t^m}$ and $M_a$ .) As $M_a$ represents $L_a\in \text{End}_B(R/Rf)$ , $(B^k,\ast )$ is isomorphic to $R/Rf$ equipped with the multiplication $(a+Rf)(b+Rf)=L_{a(t)+\nu \rho (a_0)t^m}(b+Rf)$ . Thus, $(R_m,\circ )$ and $(B^k,\ast )$ are isomorphic algebras and $S_{n,m,1}(\nu,\rho, f)$ is the same algebra as $(R_m, \circ )$ .

If $l=1$ , we write $S(\nu,\rho, f)=S_{n,m,1}(\nu,\rho, f)$ for $(R_m,\circ )$ . $S(\nu,\rho, f)$ is a division algebra if and only if every matrix in $\mathcal{C}_{n,m,1}$ has full column rank. It then canonically defines an $F'$ -linear $MRD$ code in $M_k(B)$ , $B=\textrm{Nuc}_r(S_f)$ . Therefore, we obtain from all of the above results:

The case $\nu =0$ produces the MRD codes which are associated with the unital Petit algebras. They can be viewed as generalized Gabidulin codes.

More generally, we can also construct MRD codes for $l\gt 1$ . Let $f\in R$ not be right-invariant, and let $l\lt k$ be a positive integer.

We are not able to say if the assumption on $P$ can be satisfied in this general setup. It is satisfied in the case considered in [Reference Sheekey34, Theorem 7].

Proof. We have to show that the minimum column rank of the matrix corresponding to a nonzero element in $S_{n,m,l}(\nu,\rho, f)$ is $k-l+1$ . By Theorem 6, this is equivalent to finding an element $g\in A$ such that the greatest common right divisor of $g$ and $h$ has degree at most $(l-1)m$ . Suppose towards a contradiction that there exists $g\in A$ such that $\textrm{deg}(\textrm{gcrd}(g,h))=lm$ ; since $\textrm{deg}(g)\leq lm$ , it follows that $g$ must be a divisor of $h$ . As any divisor of $h$ is a product of irreducible polynomials similar to $f$ , $g$ must be a product of polynomials similar to $f$ . This contradicts our assumption, so any matrix has rank at least $k-l+1$ .

The proof is straightforward.

6.2. The case R = K[t; $\sigma$ ]

Let $f=\sum _{i=0}^ma_it^i\in R=K[t;\;\sigma ]$ be an irreducible monic polynomial of degree $m$ with minimal central left multiple $h$ . Suppose throughout this section that $F/F'$ is a finite field extension, $\nu \in K$ , and $\rho \in \textrm{Aut}(K)$ . Let $1\lt l\lt k$ and $S_{n,m,l}(\nu,\rho, f)=\lbrace a+Rh\,|\, a\in P\rbrace \subset R/Rh,$ where $P=\lbrace d_0+d_1t+\dots +d_{lm-1}t^{lm-1}+\nu \rho (d_0)t^{lm}\,|\, d_i\in K \rbrace .$ Then, we obtain the following results:

The proof is analogous to the one of Theorem 15. Note that $f$ may be right-invariant here. Using Theorems 10 and 13, we obtain (for finite fields, cf. [Reference Sheekey34], the proof is analogous):

Theorem 22. Suppose that $\textrm{deg}(h)=mn$ . Then, $S(\nu,\rho, f)$ is a division algebra over $F'$ if

\begin{equation*}N_{K/F'}(a_0) N_{K/F'}(\nu )\not =1.\end{equation*}

Note that the condition on $f$ in (iii) is satisfied for all $f$ if $gcd(m,n)=1$ or if $n$ is prime.

We now look at the case that $1\lt l\lt k$ and also assume that $f$ is not right-invariant.

Theorem 24 (for finite fields, cf. [Reference Sheekey34, Theorem 7]). If $\textrm{deg}(h)=mn$ , then the set $S_{n,m,l}(\nu,\rho, f)$ defines an $F'$ -linear MRD code in $M_n(E_{\hat{h}})$ with minimum distance $n-l+1$ for any $\nu \in K$ such that

\begin{equation*}N_{K/F'}(\nu )\neq 1/N_{K/F'}(a_0)^l.\end{equation*}

Note that $k=n$ here since $\textrm{deg}(h)=mn$ .

The codes $S_{n,m,l}(0,\rho, h)$ generalize the Gabidulin codes constructed in [Reference Gabidulin10] that go back to [Reference Delsarte8].

Note that $N_{K/F'}(\nu )\not \in ({F'}^\times )^l$ implies $N_{K/F'}(\nu )\not =N_{K/F'}(a_0)^l$ for any $f$ . Thus, (ii) implies (iii) above.

Proof. We have to show that the minimum column rank of the matrix corresponding to a nonzero element in $S_{n,m,l}(\nu,\rho, f)$ is $k-l+1$ . By Theorem 7, this is equivalent to finding an element $g\in P$ such that the greatest common right divisor of $g$ and $h$ has degree at most $(l-1)m$ . Suppose towards a contradiction that $\text{deg}(gcrd(g,h))=lm$ ; since $\text{deg}(g)\leq lm$ , it follows that $g$ must be a divisor of $h$ .

As any divisor of $h$ is a product of irreducible polynomials similar to $f$ , $g$ must be a product of polynomials similar to $f$ . This contradicts our assumption, so any matrix has rank at least $k-l+1$ .