Nonassociative Petit Algebras

• Nonassociative Petit algebras are defined by right reduction modulo a polynomial f in an Ore extension, generalizing associative quotients.
• They provide a unified framework for nonassociative cyclic algebras, finite semifields, eigenring constructions, and skew-polynomial codes.
• The structure and division properties of these algebras are dictated by the irreducibility of f and the interplay of skew derivations and automorphisms.

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 (Brown, 2018, Brown et al., 2017, Pumpluen, 2015).

1. Construction from Ore extensions

Let $D$ be an associative division ring, $\sigma:D\to D$ an endomorphism, and $f$0 a left $f$1-derivation, so

$f$2

The Ore extension

$f$3

is the associative ring whose multiplication is determined by

$f$4

Special cases are $f$5 when $f$6, $f$7 when $f$8, and the ordinary polynomial ring $f$9 when both are trivial. Since $\deg(f)=m$0 is a division ring, $\deg(f)=m$1 is a left principal ideal domain and admits a right division algorithm: for $\deg(f)=m$2 and any $\deg(f)=m$3, there exist unique $\deg(f)=m$4 with $\deg(f)=m$5 such that $\deg(f)=m$6 (Pumpluen, 2015, Brown, 2018).

Fix $\deg(f)=m$7 with $\deg(f)=m$8. The additive group underlying the Petit algebra is

$\deg(f)=m$9

which is a free left $R_m=\{g\in R\mid \deg g<m\}$0-module with basis $R_m=\{g\in R\mid \deg g<m\}$1. Multiplication is defined by multiplying in $R_m=\{g\in R\mid \deg g<m\}$2 and then reducing on the right modulo $R_m=\{g\in R\mid \deg g<m\}$3: $R_m=\{g\in R\mid \deg g<m\}$4 The constant polynomial $R_m=\{g\in R\mid \deg g<m\}$5 is a two-sided identity, so $R_m=\{g\in R\mid \deg g<m\}$6 is a unital algebra. As an additive group, it is identified with $R_m=\{g\in R\mid \deg g<m\}$7, but the multiplication is induced by right remainder rather than by an associative quotient unless $R_m=\{g\in R\mid \deg g<m\}$8 is two-sided (Pumpluen, 2015, Brown et al., 2017).

In the twisted case $R_m=\{g\in R\mid \deg g<m\}$9 and for

$g\circ h=(gh)\bmod_r f.$0

the multiplication is explicit on the monomial basis: $g\circ h=(gh)\bmod_r f.$1 This formula is the standard model for skew-cyclic and skew-constacyclic constructions (Pumpluen, 2015).

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

$g\circ h=(gh)\bmod_r f.$2

where $g\circ h=(gh)\bmod_r f.$3 is the center of $g\circ h=(gh)\bmod_r f.$4. In the field case $g\circ h=(gh)\bmod_r f.$5, Petit’s scalar field is

$g\circ h=(gh)\bmod_r f.$6

and when

$g\circ h=(gh)\bmod_r f.$7

one has $g\circ h=(gh)\bmod_r f.$8 (Brown, 2018, Brown et al., 2017).

The basic associativity criterion is sharp. If $g\circ h=(gh)\bmod_r f.$9 is right invariant, equivalently $S_f=(R_m,\circ)$0 is a two-sided ideal, then $S_f=(R_m,\circ)$1 is exactly the associative quotient $S_f=(R_m,\circ)$2. If $S_f=(R_m,\circ)$3 is not right invariant, $S_f=(R_m,\circ)$4 is nonassociative. In the twisted field case the same condition is phrased by saying that $S_f=(R_m,\circ)$5 is invariant (Brown, 2018, Brown et al., 2017).

For non-right-invariant $S_f=(R_m,\circ)$6 of degree at least $S_f=(R_m,\circ)$7, the nuclei have a particularly rigid form: $S_f=(R_m,\circ)$8 The right nucleus is the eigenring $S_f=(R_m,\circ)$9, and if $R/Rf$0 is irreducible then $R/Rf$1 is an associative division ring. In the same non-right-invariant situation the center is

$R/Rf$2

When $R/Rf$3 has $R/Rf$4, one has $R/Rf$5, hence

$R/Rf$6

and this subalgebra is a field if $R/Rf$7 is irreducible in $R/Rf$8 (Brown, 2018, Owen, 2022).

The thesis on the right nucleus refines this picture in the bounded case. If $R/Rf$9 is bounded and its minimal central left multiple is

$f$0

with $f$1 irreducible, then $f$2 is a central simple algebra over

$f$3

In the twisted field case $f$4, the degree of $f$5 divides $f$6, where $f$7 is the order of $f$8. Over finite fields this implies that for irreducible $f$9, the right nucleus is a field of dimension $D$0 over the base field, recovering the usual size formula for cyclic Petit semifields (Owen, 2022).

A further structural invariant is the set of semi-invariant coefficients

$D$1

If $D$2 is not right invariant, then

$D$3

so the full nucleus inside the coefficient ring is exactly the set of semi-invariant elements (Owen, 2022).

3. Irreducibility, division algebras, and cyclic forms

The irreducibility of $D$4 governs the division property. Brown proves that for $D$5,

$D$6

If, in addition, $D$7 is finite-dimensional over $D$8 or is a free right $D$9-module of finite rank, then

$\sigma:D\to D$0

In particular, when $\sigma:D\to D$1 is a finite field, $\sigma:D\to D$2, and $\sigma:D\to D$3 is irreducible, $\sigma:D\to D$4 is a finite semifield. Brown records the Lavrauw–Sheekey statement that every Jha–Johnson semifield is isotopic to some $\sigma:D\to D$5 (Brown, 2018).

A central special family is

$\sigma:D\to D$6

In the finite-field coding context,

$\sigma:D\to D$7

Thus the same polynomial can produce either the associative quotient $\sigma:D\to D$8 or a genuinely nonassociative Petit algebra, depending on whether $\sigma:D\to D$9 lies in the fixed field and whether $f$00 divides the order of $f$01 (Pumpluen, 2015).

When $f$02 is a cyclic Galois extension of degree $f$03 with $f$04, the choice

$f$05

yields the nonassociative cyclic algebra

$f$06

If $f$07 are linearly independent over $f$08, then this algebra is a division algebra; in particular, if $f$09 is prime, it is a division algebra for every $f$10 (Brown et al., 2017, Brown, 2018).

For reducibility, the note on codes records concrete criteria. For example, in $f$11,

$f$12

is reducible if and only if there exists $f$13 such that

$f$14

and $f$15 is always reducible. If $f$16 is prime and the fixed field contains a primitive $f$17-th root of unity, then $f$18 is reducible if and only if there exists $f$19 with

$f$20

These criteria explain why irreducible $f$21 are natural for semifields, whereas reducible $f$22 are natural for code constructions (Pumpluen, 2015).

4. Automorphisms, isomorphisms, and isotopy

For $f$23, $f$24 of order $f$25, and

$f$26

monic and not invariant, the automorphism theory is explicit under the hypotheses that $f$27 commutes with all $f$28-automorphisms of $f$29 and $f$30. Then every $f$31-automorphism of $f$32 is of the form

$f$33

where $f$34, $f$35, and

$f$36

In particular,

$f$37

For $f$38, this reduces to the norm condition

$f$39

and in the cyclic Galois case automorphisms fixing $f$40 are inner and correspond to $f$41 (Brown et al., 2017, Brown, 2018).

The same coefficient relations classify isomorphisms. If

$f$42

are monic, not invariant, and $f$43, then

$f$44

if and only if there exist $f$45 and $f$46 such that

$f$47

In particular, the zero-pattern of the coefficients is preserved by isomorphism (Brown et al., 2017).

A later refinement concerns isotopy classes of Petit division algebras. For $f$48 with $f$49 cyclic Galois, two irreducible skew polynomials $f$50 are similar if and only if they have the same bound, equivalently the same minimal central left multiple. If $f$51 and $f$52 are similar and irreducible, then the Petit division algebras $f$53 and $f$54 are isotopic. More generally, if the irreducible polynomials $f$55 defining the minimal central left multiples of $f$56 and $f$57 have the same degree and lie in the same orbit of the group

$f$58

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 $f$59-orbits on monic irreducible polynomials in $f$60 (Pumpluen, 23 Nov 2025).

5. Left ideals and the coding-theoretic interpretation

A decisive structural fact is that left ideals of $f$61 are controlled by right divisors of $f$62. If $f$63, then every left ideal in $f$64 is generated by a monic right divisor $f$65 of $f$66. If $f$67 is irreducible, $f$68 has no nontrivial left ideals (Pumpluen, 2015).

When $f$69, $f$70, and $f$71 is monic of degree $f$72, each element

$f$73

corresponds to a vector $f$74. A left ideal $f$75 therefore defines a linear code

$f$76

The note on coding shows that cyclic submodules, $f$77-codes, module $f$78-codes, skew-cyclic codes, and skew-constacyclic codes all fit this single left-ideal description (Pumpluen, 2015).

For

$f$79

left multiplication by $f$80 realizes the skew constacyclic shift: $f$81 Hence

$f$82

is the induced action on coefficient vectors. The paper proves that a linear code $f$83 is $f$84-constacyclic with constant $f$85 if and only if its skew-polynomial representation is a left ideal of $f$86, generated by a monic right divisor of $f$87. The specialization $f$88 gives skew-cyclic codes (Pumpluen, 2015).

The conceptual shift is that two-sidedness of $f$89 is no longer required. Classical quotient-ring constructions $f$90 force $f$91 to be two-sided, hence impose conditions such as $f$92. The Petit-algebra formulation works for arbitrary monic $f$93, including non-two-sided $f$94, because $f$95 exists as a unital nonassociative algebra whether or not $f$96 is two-sided (Pumpluen, 2015).

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

$f$97

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 (Brown, 2018).

A recent extension is provided by Menichetti’s nonassociative $f$98-crossed products. For an abelian Galois extension $f$99 with $\deg(f)=m$00, the algebra

$\deg(f)=m$01

is a unital central $\deg(f)=m$02-algebra of dimension $\deg(f)=m$03 whose nucleus contains $\deg(f)=m$04. In the cyclic case,

$\deg(f)=m$05

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 $\deg(f)=m$06 (Pumpluen, 2024).

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

$\deg(f)=m$07

with a central simple algebra $\deg(f)=m$08 in place of the nucleus field, and situates them in the semiassociative Brauer monoid (Pumpluen, 2024).

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 $\deg(f)=m$09 (Brown, 2018, Pumpluen, 2015, Pumpluen, 2024).