---
title: Nonassociative Petit Algebras
url: https://www.emergentmind.com/topics/nonassociative-petit-algebras
type: topic
---

# Nonassociative Petit Algebras

Nonassociative Petit algebras are unital algebras obtained from a skew-polynomial ring \(R=D[t;\sigma,\delta]\) by replacing ordinary quotient multiplication with right reduction modulo a polynomial \(f\). If \(\deg(f)=m\), one sets \(R_m=\{g\in R\mid \deg g<m\}\) and defines
\[
g\circ h=(gh)\bmod_r f.
\]
The resulting algebra \(S_f=(R_m,\circ)\) generalizes the associative quotient \(R/Rf\): it is associative precisely when \(f\) is right invariant (equivalently, invariant or two-sided in the field case), and it is genuinely nonassociative otherwise. Introduced by Petit in the 1960s, these algebras form a common framework for nonassociative cyclic algebras, finite semifields, eigenring constructions, and several skew-polynomial code families [1806.00822][1703.00718][1504.00190].

## 1. Construction from Ore extensions

Let \(D\) be an associative division ring, \(\sigma:D\to D\) an endomorphism, and \(\delta:D\to D\) a left \(\sigma\)-derivation, so
\[
\delta(ab)=\sigma(a)\delta(b)+\delta(a)b.
\]
The Ore extension
\[
R=D[t;\sigma,\delta]
\]
is the associative ring whose multiplication is determined by
\[
ta=\sigma(a)t+\delta(a).
\]
Special cases are \(D[t;\sigma]\) when \(\delta=0\), \(D[t;\delta]\) when \(\sigma=\mathrm{id}\), and the ordinary polynomial ring \(D[t]\) when both are trivial. Since \(D\) is a division ring, \(R\) is a left principal ideal domain and admits a right division algorithm: for \(f\neq 0\) and any \(g\in R\), there exist unique \(q,r\in R\) with \(\deg r<\deg f\) such that \(g=qf+r\) [1504.00190][1806.00822].

Fix \(f\in R\) with \(\deg f=m\ge 2\). The additive group underlying the Petit algebra is
\[
R_m=\{g\in R\mid \deg g<m\},
\]
which is a free left \(D\)-module with basis \(\{1,t,\dots,t^{m-1}\}\). Multiplication is defined by multiplying in \(R\) and then reducing on the right modulo \(f\):
\[
g\circ h=gh\bmod_r f.
\]
The constant polynomial \(1\) is a two-sided identity, so \(S_f=(R_m,\circ)\) is a unital algebra. As an additive group, it is identified with \(R/Rf\), but the multiplication is induced by right remainder rather than by an associative quotient unless \(Rf\) is two-sided [1504.00190][1703.00718].

In the twisted case \(R=D[t;\sigma]\) and for
\[
f(t)=t^m-d,
\]
the multiplication is explicit on the monomial basis:
\[
(a t^i)\circ (b t^j)=
\begin{cases}
a\,\sigma^i(b)\, t^{i+j}, & i+j<m,\\[0.3em]
a\,\sigma^i(b)\, t^{i+j-m}d, & i+j\ge m,
\end{cases}
\qquad a,b\in D.
\]
This formula is the standard model for skew-cyclic and skew-constacyclic constructions [1504.00190].

## 2. Associativity, scalar field, and nuclei

The scalar field of a Petit algebra is described in two closely related ways. In the general Ore-extension setting Brown writes
\[
F=C\cap \mathrm{Fix}(\sigma)\cap \mathrm{Const}(\delta),
\]
where \(C\) is the center of \(D\). In the field case \(R=K[t;\sigma]\), Petit’s scalar field is
\[
F_0=\{a\in K\mid ah=ha\text{ for all }h\in S_f\},
\]
and when
\[
f(t)=t^m-\sum_{i=0}^{m-1} a_i t^i,\qquad a_0\neq 0,
\]
one has \(F_0=F=\mathrm{Fix}(\sigma)\) [1806.00822][1703.00718].

The basic associativity criterion is sharp. If \(f\) is right invariant, equivalently \(Rf\) is a two-sided ideal, then \(S_f\) is exactly the associative quotient \(R/Rf\). If \(f\) is not right invariant, \(S_f\) is nonassociative. In the twisted field case the same condition is phrased by saying that \(f\) is invariant [1806.00822][1703.00718].

For non-right-invariant \(f\) of degree at least \(2\), the nuclei have a particularly rigid form:
\[
\mathrm{Nuc}_\ell(S_f)=\mathrm{Nuc}_m(S_f)=D,
\qquad
\mathrm{Nuc}_r(S_f)=\{g\in R_m\mid fg\in Rf\}.
\]
The right nucleus is the eigenring \(\mathcal E(f)\), and if \(f\) is irreducible then \(\mathcal E(f)\) is an associative division ring. In the same non-right-invariant situation the center is
\[
\mathrm{Cent}(S_f)=F=C\cap\mathrm{Fix}(\sigma)\cap\mathrm{Const}(\delta).
\]
When \(f(t)=t^m-\sum_{i=0}^{m-1} a_i t^i\) has \(a_i\in F\), one has \(t\in\mathrm{Nuc}_r(S_f)\), hence
\[
F[t]/F[t]f(t)\subseteq \mathrm{Nuc}_r(S_f),
\]
and this subalgebra is a field if \(f\) is irreducible in \(F[t]\) [1806.00822][2206.09436].

The thesis on the right nucleus refines this picture in the bounded case. If \(f\) is bounded and its minimal central left multiple is
\[
h(t)=\hat h(\cdots)
\]
with \(\hat h\) irreducible, then \(\mathrm{Nuc}_r(S_f)\) is a central simple algebra over
\[
E_{\hat h}=F[x]/(\hat h(x)).
\]
In the twisted field case \(K[t;\sigma]\), the degree of \(\mathrm{Nuc}_r(S_f)\) divides \(\gcd(m,n)\), where \(n\) is the order of \(\sigma\). Over finite fields this implies that for irreducible \(f\), the right nucleus is a field of dimension \(m\) over the base field, recovering the usual size formula for cyclic Petit semifields [2206.09436].

A further structural invariant is the set of semi-invariant coefficients
\[
L^{(\sigma,\delta,f)}=\{c\in D\mid fc\in Df\}.
\]
If \(f\) is not right invariant, then
\[
L^{(\sigma,\delta,f)}=\mathrm{Nuc}(S_f)=D\cap \mathrm{Nuc}_r(S_f),
\]
so the full nucleus inside the coefficient ring is exactly the set of semi-invariant elements [2206.09436].

## 3. Irreducibility, division algebras, and cyclic forms

The irreducibility of \(f\) governs the division property. Brown proves that for \(R=D[t;\sigma,\delta]\),
\[
f\text{ irreducible } \Longleftrightarrow S_f\text{ is a right division algebra}.
\]
If, in addition, \(S_f\) is finite-dimensional over \(F\) or is a free right \(\mathrm{Nuc}_r(S_f)\)-module of finite rank, then
\[
S_f\text{ is a division algebra } \Longleftrightarrow f\text{ is irreducible}.
\]
In particular, when \(K\) is a finite field, \(\sigma\in\mathrm{Aut}(K)\), and \(f\in K[t;\sigma]\) is irreducible, \(S_f\) is a finite semifield. Brown records the Lavrauw–Sheekey statement that every Jha–Johnson semifield is isotopic to some \(S_f\) [1806.00822].

A central special family is
\[
f(t)=t^m-d\in K[t;\sigma].
\]
In the finite-field coding context,
\[
f(t)=t^m-d\text{ is two-sided } \iff m\mid \mathrm{ord}(\sigma)\text{ and } d\in \mathrm{Fix}(\sigma).
\]
Thus the same polynomial can produce either the associative quotient \(R/(f)\) or a genuinely nonassociative Petit algebra, depending on whether \(d\) lies in the fixed field and whether \(m\) divides the order of \(\sigma\) [1504.00190].

When \(K/F\) is a cyclic Galois extension of degree \(m\) with \(\mathrm{Gal}(K/F)=\langle \sigma\rangle\), the choice
\[
R=K[t;\sigma^{-1}],\qquad f(t)=t^m-d,\qquad d\notin F
\]
yields the nonassociative cyclic algebra
\[
(K/F,\sigma,d)=S_f.
\]
If \(1,d,\dots,d^{m-1}\) are linearly independent over \(F\), then this algebra is a division algebra; in particular, if \(m\) is prime, it is a division algebra for every \(d\in K\setminus F\) [1703.00718][1806.00822].

For reducibility, the note on codes records concrete criteria. For example, in \(K[t;\sigma]\),
\[
t^3-d
\]
is reducible if and only if there exists \(z\in K\) such that
\[
\sigma(z)\sigma^2(z)z=d
\quad\text{or}\quad
\sigma(d)=\sigma^2(z)\sigma(z)z,
\]
and \(t^3-1\) is always reducible. If \(m\) is prime and the fixed field contains a primitive \(m\)-th root of unity, then \(t^m-d\) is reducible if and only if there exists \(z\in K\) with
\[
d=\sigma^{m-1}(z)\cdots \sigma(z)z
\quad\text{or}\quad
\sigma^{m-1}(d)=\sigma^{m-1}(z)\cdots \sigma(z)z.
\]
These criteria explain why irreducible \(f\) are natural for semifields, whereas reducible \(f\) are natural for code constructions [1504.00190].

## 4. Automorphisms, isomorphisms, and isotopy

For \(R=K[t;\sigma]\), \(\sigma\in\mathrm{Aut}(K)\) of order \(n\), and
\[
f(t)=t^m-\sum_{i=0}^{m-1} a_i t^i
\]
monic and not invariant, the automorphism theory is explicit under the hypotheses that \(\sigma\) commutes with all \(F\)-automorphisms of \(K\) and \(n\ge m-1\). Then every \(F\)-automorphism of \(S_f\) is of the form
\[
H_{T,k}\Bigl(\sum_{i=0}^{m-1} x_i t^i\Bigr)
=
T(x_0)+\sum_{i=1}^{m-1} T(x_i)\Bigl(\prod_{l=0}^{i-1}\sigma^l(k)\Bigr)t^i,
\]
where \(T\in\mathrm{Aut}_F(K)\), \(k\in K^\times\), and
\[
T(a_i)=\Bigl(\prod_{l=i}^{m-1}\sigma^l(k)\Bigr)a_i
\quad\text{for all }i.
\]
In particular,
\[
H_{T,k}(t)=kt.
\]
For \(f(t)=t^m-a\), this reduces to the norm condition
\[
T(a)=\Bigl(\prod_{l=0}^{m-1}\sigma^l(k)\Bigr)a,
\]
and in the cyclic Galois case automorphisms fixing \(K\) are inner and correspond to \(\ker N_{K/F}\) [1703.00718][1806.00822].

The same coefficient relations classify isomorphisms. If
\[
f(t)=t^m-\sum_{i=0}^{m-1} a_i t^i,\qquad
g(t)=t^m-\sum_{i=0}^{m-1} b_i t^i
\]
are monic, not invariant, and \(n\ge m-1\), then
\[
S_f\cong_F S_g
\]
if and only if there exist \(T\in\mathrm{Aut}_F(K)\) and \(k\in K^\times\) such that
\[
T(a_i)=\Bigl(\prod_{l=i}^{m-1}\sigma^l(k)\Bigr)b_i
\quad\text{for all }i.
\]
In particular, the zero-pattern of the coefficients is preserved by isomorphism [1703.00718].

A later refinement concerns isotopy classes of Petit division algebras. For \(R=K[t;\sigma]\) with \(K/F\) cyclic Galois, two irreducible skew polynomials \(f,g\) are similar if and only if they have the same bound, equivalently the same minimal central left multiple. If \(f\) and \(g\) are similar and irreducible, then the Petit division algebras \(R/Rf\) and \(R/Rg\) are isotopic. More generally, if the irreducible polynomials \(\hat h_1,\hat h_2\in F[x]\) defining the minimal central left multiples of \(f\) and \(g\) have the same degree and lie in the same orbit of the group
\[
G=N_{K/F}(K^\times)\rtimes \mathrm{Aut}(K)_\sigma,
\]
then the corresponding Petit division algebras are isotopic. In finite fields, this leads to an explicit upper bound for the number of isotopy classes in terms of \(G\)-orbits on monic irreducible polynomials in \(F[x]\) [2511.18451].

## 5. Left ideals and the coding-theoretic interpretation

A decisive structural fact is that left ideals of \(S_f\) are controlled by right divisors of \(f\). If \(f\in R=D[t;\sigma,\delta]\), then every left ideal in \(S_f\) is generated by a monic right divisor \(g\) of \(f\). If \(f\) is irreducible, \(S_f\) has no nontrivial left ideals [1504.00190].

When \(K=\mathbb F_q\), \(\sigma\in\mathrm{Aut}(K)\), and \(f\) is monic of degree \(m\), each element
\[
a(t)=a_0+a_1t+\cdots+a_{m-1}t^{m-1}\in S_f
\]
corresponds to a vector \((a_0,\dots,a_{m-1})\in K^m\). A left ideal \(I\subseteq S_f\) therefore defines a linear code
\[
C_I=\{(a_0,\dots,a_{m-1})\mid a(t)\in I\}.
\]
The note on coding shows that cyclic submodules, \((\sigma,\delta)\)-codes, module \(\sigma\)-codes, skew-cyclic codes, and skew-constacyclic codes all fit this single left-ideal description [1504.00190].

For
\[
f(t)=t^m-d\in K[t;\sigma],
\]
left multiplication by \(t\) realizes the skew constacyclic shift:
\[
t a(t)=\sigma(a_{m-1})d+\sigma(a_0)t+\cdots+\sigma(a_{m-2})t^{m-1}.
\]
Hence
\[
(a_0,\dots,a_{m-1})
\mapsto
(\sigma(a_{m-1})d,\sigma(a_0),\dots,\sigma(a_{m-2}))
\]
is the induced action on coefficient vectors. The paper proves that a linear code \(C\subseteq K^m\) is \(\sigma\)-constacyclic with constant \(d\) if and only if its skew-polynomial representation is a left ideal of \(S_f\), generated by a monic right divisor of \(t^m-d\). The specialization \(d=1\) gives skew-cyclic codes [1504.00190].

The conceptual shift is that two-sidedness of \(f\) is no longer required. Classical quotient-ring constructions \(R/(t^m-1)\) force \(t^m-1\) to be two-sided, hence impose conditions such as \(m\mid \mathrm{ord}(\sigma)\). The Petit-algebra formulation works for arbitrary monic \(f\), including non-two-sided \(t^m-d\), because \(S_f\) exists as a unital nonassociative algebra whether or not \(Rf\) is two-sided [1504.00190].

## 6. Broader generalizations and historical setting

Petit’s original goal was the construction of quasi-fields and semifields from skew-polynomial rings. Later work made clear that these algebras also organize several older crossed-product constructions. Brown’s thesis shows that generalized cyclic algebras of the form
\[
A_i[t_i;\tau_i]/A_i[t_i;\tau_i](t_i^{q_i}-c_i)
\]
can be assembled into chains, and a central simple algebra is a solvable crossed product if and only if it can be written as such a chain of generalized cyclic algebras; this places Petit-type constructions inside the structure theory of solvable crossed products [1806.00822].

A recent extension is provided by Menichetti’s nonassociative \(G\)-crossed products. For an abelian Galois extension \(K/F\) with \(\mathrm{Gal}(K/F)=G\), the algebra
\[
(K/F,k_0,\dots,k_{m-1})
\]
is a unital central \(F\)-algebra of dimension \(m^2\) whose nucleus contains \(K\). In the cyclic case,
\[
(K/F,1,\dots,1,k)^{\mathrm{op}}\cong (K/F,\sigma,k),
\]
so the construction specializes to the opposite algebra of a classical nonassociative cyclic algebra of Petit type. More generally, Menichetti algebras are described as special cases of Albert’s nonassociative crossed extensions, which include the generalized Petit-type quotients \(K[t;\theta]/K[t;\theta](t^m-d)\) [2407.16256].

These crossed-product generalizations retain several characteristic Petit features: a large associative nucleus, centrality over the base field, explicit matrix models for multiplication, and division criteria formulated through determinant or norm-type conditions. The same paper extends the construction further to algebras
\[
(D,\varphi,k_0,\dots,k_{m-1})
\]
with a central simple algebra \(D\) in place of the nucleus field, and situates them in the semiassociative Brauer monoid [2407.16256].

Historically, the subject thus runs from Petit’s 1960s semifield construction through nonassociative cyclic algebras, skew-polynomial code theory, automorphism and isotopy classifications, and nonassociative crossed products. Across these settings, the defining mechanism remains the same: a skew-polynomial multiplication in an Ore extension, followed by right reduction modulo a chosen polynomial \(f\) [1806.00822][1504.00190][2407.16256].

Source: https://www.emergentmind.com/topics/nonassociative-petit-algebras