A maximal T-space of F₃[x]₀
In earlier work, we have established that for any finite field k, the free associative k-algebra on one generator x, denoted by k[x]₀, has infinitely many maximal T-spaces, but exactly two maximal T-ideals (each of which is a maximal T-space). However, aside from these two T-ideals, no specific exam...
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | A maximal T-space of F₃[x]₀ / C. Bekh-Ochir, S.A. Rankin // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 160–170. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1523432019-06-11T01:25:03Z A maximal T-space of F₃[x]₀ Bekh-Ochir, C. Rankin, S.A. In earlier work, we have established that for any finite field k, the free associative k-algebra on one generator x, denoted by k[x]₀, has infinitely many maximal T-spaces, but exactly two maximal T-ideals (each of which is a maximal T-space). However, aside from these two T-ideals, no specific examples of maximal T-spaces of k[x]₀ were determined at that time. In a subsequent work, we proposed that for a finite field k of characteristic p > 2 and order q, for each positive integer n which is a power of 2, the T-space Wn, generated by {x + xqⁿ, xqⁿ⁺¹}, is maximal, and we proved that W₁ is maximal. In this note, we prove that for q = p = 3, W₂ is maximal. 2013 Article A maximal T-space of F₃[x]₀ / C. Bekh-Ochir, S.A. Rankin // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 160–170. — Бібліогр.: 5 назв. — англ. 1726-3255 2010 MSC:16R10. http://dspace.nbuv.gov.ua/handle/123456789/152343 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In earlier work, we have established that for any finite field k, the free associative k-algebra on one generator x, denoted by k[x]₀, has infinitely many maximal T-spaces, but exactly two maximal T-ideals (each of which is a maximal T-space). However, aside from these two T-ideals, no specific examples of maximal T-spaces of k[x]₀ were determined at that time. In a subsequent work, we proposed that for a finite field k of characteristic p > 2 and order q, for each positive integer n which is a power of 2, the T-space Wn, generated by {x + xqⁿ, xqⁿ⁺¹}, is maximal, and we proved that W₁ is maximal. In this note, we prove that for q = p = 3, W₂ is maximal. |
| format |
Article |
| author |
Bekh-Ochir, C. Rankin, S.A. |
| spellingShingle |
Bekh-Ochir, C. Rankin, S.A. A maximal T-space of F₃[x]₀ Algebra and Discrete Mathematics |
| author_facet |
Bekh-Ochir, C. Rankin, S.A. |
| author_sort |
Bekh-Ochir, C. |
| title |
A maximal T-space of F₃[x]₀ |
| title_short |
A maximal T-space of F₃[x]₀ |
| title_full |
A maximal T-space of F₃[x]₀ |
| title_fullStr |
A maximal T-space of F₃[x]₀ |
| title_full_unstemmed |
A maximal T-space of F₃[x]₀ |
| title_sort |
maximal t-space of f₃[x]₀ |
| publisher |
Інститут прикладної математики і механіки НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/152343 |
| citation_txt |
A maximal T-space of F₃[x]₀ / C. Bekh-Ochir, S.A. Rankin // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 2. — С. 160–170. — Бібліогр.: 5 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT bekhochirc amaximaltspaceoff3x0 AT rankinsa amaximaltspaceoff3x0 AT bekhochirc maximaltspaceoff3x0 AT rankinsa maximaltspaceoff3x0 |
| first_indexed |
2025-07-13T02:52:01Z |
| last_indexed |
2025-07-13T02:52:01Z |
| _version_ |
1837498499558539264 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 16 (2013). Number 2. pp. 160 – 170
c© Journal “Algebra and Discrete Mathematics”
A maximal T -space of F3[x]0
Chuluun Bekh-Ochir and Stuart A. Rankin
Communicated by V. A. Artamonov
Abstract. In earlier work, we have established that for
any finite field k, the free associative k-algebra on one generator x,
denoted by k[x]0, has infinitely many maximal T -spaces, but exactly
two maximal T -ideals (each of which is a maximal T -space). However,
aside from these two T -ideals, no specific examples of maximal T -
spaces of k[x]0 were determined at that time. In a subsequent work,
we proposed that for a finite field k of characteristic p > 2 and order
q, for each positive integer n which is a power of 2, the T -space Wn,
generated by { x + xq
n
, xq
n
+1 }, is maximal, and we proved that
W1 is maximal. In this note, we prove that for q = p = 3, W2 is
maximal.
1. Introduction
Let k be a field, and let A be an associative k-algebra. A. V. Grishin
introduced the concept of a T -space of A ([3], [4]); namely, a linear subspace
of A that is invariant under the natural action of the transformation
monoid of all k-algebra endomorphisms of A. A T -space of A that is also
an ideal of A is called a T -ideal of A. For any H ⊆ A, the smallest T -space
of A containing H shall be denoted by HS . The set of all T -spaces of A
forms a lattice under the inclusion ordering.
We shall let k[x]0 denote the free associative k-algebra on the single
generator x (so k[x]0 = xk[x]). It was shown in [1] that there is a natural
bijection between the set of maximal T -spaces of any free associative
k-algebra and the set of maximal T -spaces of k[x]0. It was also proven that
if k is infinite, then any free associative k-algebra has a unique maximal
T -ideal, and that maximum T -ideal is also the unique maximal T -space,
2010 MSC: 16R10.
C. Bekh-Ochir, S. A. Rankin 161
while when k is finite, then any free associative k-algebra has two maximal
T -ideals, each of which is a also a maximal T -space, but that now there
are infinitely many maximal T -spaces. However, no explicit examples of
maximal T -spaces, other than the two maximal T -ideals, were known. In
a subsequent paper ([2]), we proposed that for any prime p > 2, and any
finite field k of characteristic p and order q, the T -spaces Wn, where n is
any power of 2 that is greater than 1, defined by
Wn = { x + xqn
, xqn+1 }
were each maximal in k[x]0. We remark that we had also established in [2]
that for r = 2sm, with m odd, Wr ⊆ W2s , and that for r > s ≥ 0, W2s is
a proper subspace of k[x]0 while W2r + W2s = k[x]0. Further, we showed
that when p = 2, W2n = { x + xq2
n
, xq2
n
+1 }S ⊆ { x + xq, xq2
n
+1 }S , and
that x + xq /∈ W2n for n > 0. For n = 0, the two T -spaces coincide, and
we did prove that W20 is a maximal T -space of k[x]0 for any prime p. Our
conjecture then was that for p > 2, W2n is a maximal T -space of k[x]0 for
every n ≥ 0, and for p = 2, { x + xq, xq2
n
+1 }S is a maximal T -space of
k[x]0 for each n ≥ 0. It does not appear that the methods that were used
to prove that W1 is maximal extend to Wn for n ≥ 2. Our objective in
this paper is to prove that for q = p = 3, W2 is maximal in F3[x]0, thereby
lending some additional support for the conjecture. We remark that these
values were chosen simply because the computation was feasible, and even
then, it was only feasible because we were able to develop a reduction
strategy that we were unable to duplicate even in the case q = p = 5 for
example.
The following notion will be of fundamental importance in our work.
Recall that for any finite field k of order q, monomials ui ∈ k[x]0 and
αi ∈ k, 1 ≤ i ≤ t, f =
∑t
i=1 αiui are said to be q-homogeneous if for each
i, j with 1 ≤ i, j ≤ t, deg(ui) ≡ deg(uj) (mod q − 1). For anyT -space T
of k[x]0, each f ∈ T can be written as a sum of q-homogeneous elements,
each of which belongs to T . Consequently, if for a positive integer i, we
let Ti denote the linear space
Ti = { 0 } ∪ { f ∈ T − { 0 } | deg(f) ≡ i (mod q − 1) },
then T is the direct sum of T1, T2, . . . , Tq−1. In particular, if we let Hi =
(k[x]0)i, then k[x]0 is the direct sum of H1, . . . , Hq−1, and Ti = T ∩ Hi.
For each i, let πi :k[x]0 → Hi ⊆ k[x]0 denote the ith projection mapping.
Then πi(T ) = Ti ⊆ T ; that is, each T -space of k[x]0 is invariant under
each of these projection mappings.
It was established in [2] that for n ≥ 1 any power of 2, the T -ideal Un
that is generated by the set { x − xq2n
} is contained in Wn, and moreover,
162 A maximal T -space of F3[x]0
that the set { (xq2n
− x)xi | i ≥ 0 } is a linear basis for Un. As well, it was
proven that the set
{ xiqn+j + xi+jqn
| qn > i > j ≥ 0 } ∪ { (xqn+1)i | 1 ≤ i ≤ qn − 1 }
is linearly independent in k[x]0, and so as a k-vector space, Vn, the linear
span of this set in k[x]0, has dimension
(qn
2
)
+ qn − 1. It was also proven
that as linear spaces, Wn = Vn ⊕ Un, and that the set
Bn = { xi+qnj | qn > i > j ≥ 0 }
is linearly independent in k[x]0, and Yn, the linear span of Bn in k[x]0, is
complementary to Wn; that is, k[x]0 = Yn ⊕ Wn = Yn ⊕ Vn ⊕ Un. Thus
in order to establish that Wn is maximal in k[x]0, it suffices to show that
for any nonzero f ∈ Yn, Wn + { f }S = k[x]0. Moreover, by the preceding
discussion on q-homogeneity, it will suffice to prove that for any nonzero
q-homogeneous polynomial f ∈ Yn, Wn + { f }S = k[x]0.
Our general approach in this note will be to consider a T -space of
the form Wn + { f }S for nonzero f ∈ Yn. We shall take advantage of the
following observation. For θ any k-linear operator on k[x]0 = Yn ⊕ Wn
that preserves every T -space (for example, any algebra endomorphism,
or any of the projection mappings πi :k[x]0 → Hi ⊆ k[x]0), then we have
a linear operator on Yn given by
Yn →֒ Yn ⊕ Wn
θ
→ Yn ⊕ Wn
πY→ Yn,
where πY is the projection mapping onto the subspace Yn. The value to
us of this observation is the following. Let V be any T -space of k[x]0 that
contains Wn. Then for v ∈ V , we have v = y + w for unique y ∈ Yn and
w ∈ Wn. Since Wn ⊆ V , we have y = v − w ∈ V , and so y ∈ V ∩ Yn. But
then y + Wn ⊆ V , and so V = (V ∩ Yn) ⊕ Wn. Since θ preserves V , we
have θ(y) = y′ + w′ ∈ V , where y′ ∈ Yn and w′ ∈ Wn. Thus y′ ∈ V , and
so πY ◦ θ(y) = y′ ∈ V ∩ Yn; that is, πY ◦ θ(V ∩ Yn) ⊆ V ∩ Yn.
2. W2 is maximal in the case k = F3
We specialize the results described in the preceding section to the
case when k = F3 and n = 2. Since q = p = 3, q − 1 = 2 and thus
q-homogeneity is simply parity; that is, f ∈ F3[x]0 is 3-homogeneous if
and only if all of its monomials have degrees of the same parity. We have
W2 = V2 ⊕ U2, and F3[x]0 = Y2 ⊕ W2, where each of V2 and Y2 are finite
dimensional (and thus finite) linear subspaces of F3[x]0. Y2 has dimension
C. Bekh-Ochir, S. A. Rankin 163
(q2
2
)
=
(
9
2
)
= 36, with basis B2 the union of
O = { x, x3, x5, x7, x11, x13,x15, x17, x21, x23,
x25, x31, x33, x35, x41, x43, x51, x53, x61, x71 }
a set of size 20, and
E = { x2, x4, x6, x8, x12, x14, x16, x22, x24, x26, x32, x34, x42, x44, x52, x62 },
a set of size 16. We may regard πY ◦ π1 as the projection from F3[x]0 onto
〈 O 〉, and so we shall refer to πY ◦ π1 as πO for convenience. Similarly, we
shall refer to πY ◦ π2 as πE .
There are two families of algebra endomorphisms that shall be of
particular interest: those determined by mapping x to xr or x + xr for r
a positive integer. As we shall make frequent use of such endomorphisms,
we introduce notation for them.
Definition 2.1. For each positive integer r, let ǫr and ιr be the alge-
bra endomorphisms of F3[x]0 that are determined by sending x to xr,
respectively, to x + xr.
It will be convenient to note that for any T -space W that contains
W2, x80+i ≡ xi (mod W ) and x10i ≡ 0 (mod W ) for each positive integer
i, and for 9 > i > j ≥ 0, x9i+j ≡ −xi+9j (mod W ).
Proposition 2.1. W2 + { x5 }S = F3[x]0.
Proof. Let T = W2 + { x5 }5, so T is a T -space that contains W2. We
prove that x ∈ T . Since T is a T -space, ι2(T ) ⊆ T . In particular, we have
ι2(x5) = x5 − x6 + x7 + x8 − x9 + x10 ∈ T . As x5, x10 ∈ T , we have −x6 +
x7 + x8 − x9 ∈ T . The 3-homogeneous components of −x6 + x7 + x8 − x9
are −x6 + x8 and x7 − x9, so −x6 + x8 ∈ T and x7 − x9 ∈ T . As well,
x + x9 ∈ T , so x7 − x9 ≡ x7 + x (mod T ) and thus x + x7 ∈ T . But then
ι2(x + x7) = x + x2 + x7 + x8 − x10 − x11 + x13 + x14 ∈ T.
Since x + x7 ∈ T , x2 + (x2)7 = x2 + x14 ∈ T , and x10 ∈ T , we obtain that
x8 − x11 + x13 ∈ T . But then the 3-homogeneous component x8 ∈ T . We
now have that both x8 and −x6 +x8 are elements of T , and so x6 ∈ T and
thus ι2(x6) = x6 − x9 + x12 ∈ T . Since x6 and (x2)6 = x12 ∈ T , we finally
obtain x9 ∈ T . As x ≡ x9 (mod T ), it follows that x ∈ T , as required.
Let D5 =
〈
x5, x15, x25, x35
〉
. Our next major objective is to prove
that for any nonzero f ∈ D5, W2 + { f }S = F3[x]0. The fact that 5 is a
prime factor of q2n − 1 = 80 is at the heart of this observation.
The following result will prove to be useful.
164 A maximal T -space of F3[x]0
Lemma 2.1. For any α, β, γ, δ ∈ F3,
ǫ5
◦ πO ◦ ι2(αx5 + βx15 + γx25 + δx35) ≡ (α + γ + δ)x5 + (β + γ − δ)x15
+ (α + β − γ)x25 + (α − β − δ)x35 (mod W2).
Proof. We shall compute the effect of applying ǫ5
◦ πO ◦ ι2 to each of x5,
x15, x25, and x35, with each computation carried out in stages. To begin
with, we have
x5 ι27→ (x + x2)5 = (x + x2)3(x + x2)2 = (x3 + x6)(x2 + 2x3 + x4)
πO7→ x5 + x7 + 2x9 ǫ57→ x25 + x35 + 2x45
W2
≡ x5 + x25 + x35
and thus
x15 ι27→ ((x + x2)5)3 = (x9 + x18)(x6 + 2x9 + x12)
πO7→ x15 + x21 + 2x27 ǫ57→ x75 + x105 + 2x135
W2
≡ x15 + x25 − x35.
Next, we have
x25 ι27→ (x + x2)18(x + x2)6(x + x2) = (x9 + x18)2(x3 + x6)2(x + x2)
= (x18 + 2x27 + x36)(x6 + 2x9 + x12)(x + x2)
= (x18 + 2x27 + x36)(x7 + 2x10 + x13 + x8 + 2x11 + x14)
πO7→ x25 + x31 + 2x29 + x37 + 2x35 + 2x41 + x43 + x49 + 2x47
ǫ57→ x125 + x155 + 2x145 + x185 + 2x175 + 2x205 + x215 + x245 + 2x235
W2
≡ −x5 − x35 + x25 + x25 + 2x15 + x5 − x15 + x5 + x35
= x5 + x15 − x25.
Finally, we have
x35 ι27→ (x + x2)27(x + x2)6(x + x2)2
= (x27 + x54)(x6 + 2x9 + x12)(x2 + 2x3 + x4)
= (x33 + 2x36 + x39 + x60 + 2x63 + x66)(x2 + 2x3 + x4)
πO7→ x35 + x41 + 2x65 + x39 + 2x63 + 2x69 + x37 + x43 + 2x67
ǫ57→ x175 + x205 + 2x325 + x195 + 2x315 + 2x345 + x185 + x215 + 2x335
C. Bekh-Ochir, S. A. Rankin 165
W2
≡ x15 − x5 + 2x5 + x35 + x35 + 2x25 + x25 + 2x15 + 2x15
= x5 − x15 − x35.
The result follows now by linearity.
Corollary 2.1. W2 + { x15 + x25 }S = F3[x]0.
Proof. Let U = W2 + { x15 + x25 }S . We have ǫ3(x15 + x25) = x45 + x75 ≡
−x5 − x35 (mod W2), so x5 + x35 ∈ U . As well, we have ǫ5
◦ πO ◦ ι2(x15 +
x25) ∈ T , and by Lemma 2.1, ǫ5
◦ πO ◦ ι2(x15 + x25) = x5 − x15 − x35.
Since x5 + x35, x5 − x15 − x35 ∈ T , we have −(x5 + x35 + x5 − x15 −
x35) = x5 + x15 ∈ T . Thus x5 + x15 − (x15 + x25) = x5 − x25 ∈ T ,
and so ǫ3(x5 − x25) = x15 − x75 ∈ T . Since x75 ≡ −x35 (mod T ), we
then obtain that x15 + x35 ∈ T . Finally, as both x5 − x15 − x35 and
x15 +x35 belong to T , we have x5 ∈ T . Now by Proposition 2.1, we obtain
W2 + { x15 + x25 }S = F3[x]0, as required.
Proposition 2.2. For any nonzero f ∈ D5, W2 + { f }S = F3[x]0.
Proof. Let f = αx5 +βx15 +γx25 + δx35 and set T = W2 +{ f }S , so that
T is a T -space containing W2. Note that modulo W2 and thus modulo T ,
we have x45 ≡ −x5, x75 ≡ −x35, and x105 ≡ x25, while xr+80k ≡ xr for
all positive integers k, r.
We have ǫ7(f) = αx35 + βx105 + γx175 + δx245 ∈ T , and since αx35 +
βx105 + γx175 + δx245 ≡ αx35 + βx25 + γx15 + δx5, we have δx5 + γx15 +
βx25 + αx35 ∈ T . Sum f and the latter element to obtain that
(α + δ)(x5 + x35) + (β + γ)(x15 + x25) ∈ T. (1)
Apply ǫ3 to the expression in (1) to obtain that
(α + δ)(x15 + x105) + (β + γ)(x45 + x75) ∈ T (1)
and thus
(α + δ)(x15 + x25) − (β + γ)(x5 + x35) ∈ T. (2)
Multiply (1) by β + γ and (2) by α + δ and sum to obtain that
[(α + δ)2 + (β + γ)2](x15 + x25) ∈ T.
If (α + δ)2 + (β + γ)2 6= 0, then x15 + x25 ∈ T , in which case it follows
from Corollary 2.1 that T = F3[x]0. Suppose that (α + δ)2 + (β + γ)2 = 0.
Since α + δ, β + γ ∈ F3, this implies that α + δ = 0 and β + γ = 0, and so
f = αx5 + βx15 − βx25 − αx35, with not both α and β equal to 0. Apply
166 A maximal T -space of F3[x]0
Lemma 2.1 to f to obtain that
−βx5 + αx15 + (α − β)x25 − (α + β)x35 ∈ T. (3)
Add f and the expression in (3) to obtain that
(α − β)(x5 + x35) + (α + β)(x15 + x25) ∈ T. (4)
As well, we have ǫ5(f) = αx25 + βx75 − βx125 − αx175 ∈ T , and so
βx5 − αx15 + αx25 − βx35 ∈ T (5)
Add f to the expression in (5) to obtain that
(α + β)(x5 − x35) − (α − β)(x15 − x25) ∈ T. (6)
After applying ǫ3 to the expression in (6), we find that
(α + β)(x15 − x25) + (α − β)(x5 − x35) ∈ T. (7)
Now add the expressions in (4) and (7) to obtain that
(−α + β)x5 + (−α − β)x15 ∈ T, (8)
and so ǫ3((−α + β)x5 + (−α − β)x15) ∈ T . Thus
(−α + β)x15 + (α + β)x5 ∈ T. (9)
Add the expressions in (8) and (9) to find that −βx5 + αx15 ∈ T . Then
(8) together with this fact results in
αx5 + βx15 ∈ T. (10)
Apply ǫ3 to the expression in (10) to obtain that
−βx5 + αx15 ∈ T. (11)
Now multiply (10) by α, multiply (11) by β, and take the difference to
obtain that (α2 + β2)x5 ∈ T . Since α, β ∈ F3, not both zero, it follows
that α2 + β2 6= 0 and thus x5 ∈ T . Now by Proposition 2.1, we have
T = F3[x]0, as required.
Proposition 2.3. For any nonzero f ∈ 〈 O 〉, W2 + { f }S = F3[x]0.
Proof. Our approach is to prove that for any nonzero f ∈ 〈 O 〉, there
exists an algebra endomorphism θ of F3[x]0 such that πO ◦ θ(f) is a
nonzero element of D5, in which case it follows from Proposition 2.2
C. Bekh-Ochir, S. A. Rankin 167
that W2 + { f }S = F3[x]0. Since the image of the endomorphism ǫ5 is
D5, we shall regard ǫ5 as a linear map from F3[x]0 into D5. In particular,
when we apply ǫ5 to elements of 〈 O 〉, we find (using the programming
language Python) that ǫ5(
∑
xi∈O αix
i)) =
(α1 + α17 − α25 + α33 − α41)x5 + (α3 − α11 + α35 − α43 + α51)x15
+ (α5 − α13 + α21 + α53 − α61)x25 + (α7 − α15 + α23 − α31 + α71)x35.
As well, for any algebra endomorphism τ of F3[x]0, ǫ5
◦ τ can be considered
to be a linear map from F3[x]0 to D5. Let τ1 = ǫ5
◦ πO ◦ ι2, and regard
τ1 as a linear map from F3[x]0 to D5. When we apply τ1 to elements
of 〈 O 〉, we find (again, using the programming language Python) that
τ1(
∑
xi∈O αix
i)) = a1x5 + a2x15 + a3x25 + a4x35, where
a1 = α1+ α5− α17− α21− α23+ α25+ α33+ α35− α43+ α51− α53− α61+ α71
a2 = α3+ α7− α11+ α15− α17− α23+ α25− α35+ α41− α51+ α53+ α61+ α71
a3 = α5− α7+ α11− α13+ α15− α17+ α21− α25− α31
+ α43− α51− α53− α61+ α71
a4 = α5+ α7+ α13− α15− α17+ α21+ α23− α31− α33
− α35+ α41+ α51− α53− α71.
Next, let τ2 = τ1
◦ πO ◦ ι2. Again using Python, we compute that
τ2(
∑
xi∈O
αix
i) = b1x5 + b2x15 + b3x25 + b4x35,
where
b1 = α1− α5+ α11− α13− α15− α17+ α21− α31+ α33
− α35+ α41− α51− α53+ α71
b2 = α3+ α5− α7− α11− α13− α15+ α17− α25+ α31
+ α33+ α43− α51+ α53+ α71
b3 = − α11− α15− α21+ α25− α31+ α35− α41+ α51− α61+ α71
b4 = − α5− α7+ α13+ α17+ α23− α25+ α33+ α35+ α43− α53.
Let τ3 = τ2
◦ πO ◦ ι2, and compute
τ3(
∑
xi∈O
αix
i) = c1x5 + c2x15 + c3x25 + c4x35,
168 A maximal T -space of F3[x]0
where
c1 = α1+ α7− α11+ α13+ α21+ α23− α31− α33− α35− α41− α61− α71
c2 = α3− α7+ α11− α13+ α21− α23− α25− α31− α33− α43− α53− α61
c3 = α7+ α13+ α15+ α17− α23+ α31+ α33+ α35+ α41− α43− α53− α61− α71
c4 = α5+ α11− α13− α21+ α23− α25+ α31− α41− α43− α51+ α53− α61− α71.
Let τ4 = τ1
◦ πO ◦ ι3. Then τ4(
∑
xi∈O αix
i) = d1x5 + d2x15 + d3x25 + d4x35,
where
d1 = α1− α3+ α7− α11+ α15− α17− α21+ α25− α31
− α35− α41− α43− α53+ α61− α71
d2 = α1+ α3− α5+ α7− α13+ α21+ α23− α25− α33
− α35+ α41− α43− α51− α53+ α71
d3 = − α5− α7+ α13+ α15+ α17− α21+ α23+ α31+ α35+ α41+ α43− α71
d4 = α5− α7− α13+ α15+ α21− α25+ α31+ α41− α43− α51+ α53+ α61.
Finally, let τ5 = τ4
◦ ǫ2. We have τ5(
∑
xi∈O αix
i) =
(α3− α7− α17+ α23+ α33+ α43)x5 + (− α1− α11− α21− α41− α51− α61)x15
+ (α21− α31+ α61− α71)x25 + (α7+ α13− α23+ α53)x35
Each of ǫ5, τ1, τ2, τ3, τ4, and τ5 is a linear map from F3[x]0 to D5, and
each maps any T -space T containing W2 into itself. We consider the linear
map θ : 〈 O 〉 → D6
5 given by
θ(f) = (ǫ5(f), τ1(f), τ2(f), τ3(f), τ4(f), τ5(f))
for f ∈ 〈 O 〉. 〈 O 〉 has linear dimension 20, while D5 has dimension 4 and
so D6
5 has dimension 24. We used the symbolic mathematics program
SAGE ([5]) to determine that θ has rank 20, which means that θ is
injective. Thus for each nonzero f ∈ 〈 O 〉, at least one of ǫ5(f), τi(f),
i = 1, 2, 3, 4, 5, is nonzero; that is, W2 + { f }S contains a nonzero element
of D5, and thus by Proposition 2.2, W2 + { f }S = F3[x]0.
Theorem 2.1. W2 is a maximal T -space of F3[x]0.
Proof. It remains only to prove that for nonzero f ∈ 〈 E 〉, W2 + { f }S =
F3[x]0. We prove that if f ∈ 〈 E 〉 is nonzero, then W2 + { f }S contains a
nonzero element of 〈 O 〉, at which point we may apply Proposition 2.3 to
obtain that W2 + { f }S = F3[x]0.
C. Bekh-Ochir, S. A. Rankin 169
Note that if we restrict each of πO ◦ ι2 and πO ◦ ι4 to 〈 E 〉, we have
linear maps from 〈 E 〉 to 〈 O 〉, given by (again, computed with Python)
πO ◦ ι2(
∑
xi∈E
αix
i) = (α6 + α8 − α42 − α44 + α52 + α62)x
+ (−α2 + α14 + α24 + α26 − α44 − α52)x3
+ (α4 + α24 + α26 + α42 + α52)x5 + (α4 + α32 − α44 + α62)x7
+ (−α8 − α14 + α16 + α52 − α62)x11 + (−α8 + α26 − α32 + α34)x13
+ (−α8 + α12 − α14 − α44 − α62)x15
+ (α14 + α16 − α44 − α52 − α62)x17
+ (α12 + α16 + α26 − α52)x21
+ (α14 + α16 + α22 + α26 + α44 + α52)x23
+ (α14 + α16 + α22 + α34 − α52 + α62)x25
+ (α16 − α22 + α24 + α62)x31
+ (−α24 − α26 − α32 − α42 − α44)x33
+ (−α22 − α26 + α32 + α34 − α42 − α44)x35
+ (α22 + α34 + α44 − α62)x41 + (α22 − α26 − α34 + α52)x43
+ (−α26 − α32 + α42 + α44 − α52)x51
+ (α32 + α34 − α42 + α44 − α52 + α62)x61,
and πO ◦ ι4(
∑
xi∈E αix
i) =
∑
xi∈O eix
i, where
e1 = α26 + α44 − α52 − α62
e3 = −α26 + α44 + α52 − α62
e5 = −α2 + α22 + α32 − α44 − α52 − α62
e7 = α4 − α24 + α42 + α44
e11 = −α8 − α16 + α52
e13 = α4 − α16 − α34 − α44
e15 = −α6 + α16 − α26 − α34 + α44 − α52
e17 = −α8 − α14 + α52 + α62
e21 = α8 + α12 − α14 + α52
e23 = −α8 + α26 − α32 + α34 + α44 − α52
e25 = −α16 + α22 + α26 + α44 + α52 + α62
e31 = −α12 + α22 − α32 + α52
e33 = −α24 + α32 − α44
170 A maximal T -space of F3[x]0
e35 = −α26 − α32 + α34 + α44 − α52 + α62
e41 = α14 + α22 − α26 + α32
e43 = α16 − α34 + α42 + α62
e51 = −α24 + α26 − α42 − α44
e53 = −α14 + α34 + α44 + α52 + α62
e61 = α16 − α22 + α24 − α44 − α52 + α62
e71 = −α22 − α26 + α42 + α44 − α52.
Consider the linear map θ from 〈 E 〉 into 〈 O 〉2 given by
θ(f) = (πO ◦ ι2(f), πO ◦ ι4(f))
for f ∈ 〈 E 〉. 〈 E 〉 has dimension 16, and again using SAGE, we compute
that θ has rank 16, and so conclude that θ is injective. Thus for each
nonzero f ∈ 〈 E 〉, at least one of πO ◦ ι2(f) or πO ◦ ι4(f) is nonzero; that is,
W2 + { f }S contains a nonzero element of 〈 O 〉, and thus by Proposition
2.3, W2 + { f }S = F3[x]0.
References
[1] C. Bekh-Ochir, S. A. Rankin, S. A., Maximal T -spaces of a free associative algebra,
J. Algebra, 332 (2011), 442–456.
[2] C. Bekh-Ochir, S. A. Rankin, S. A., Maximal T -spaces of the free associative algebra
over a finite field, arXiv:1104.4755
[3] A. V. Grishin, On the finite-basis property of systems of generalized polynomials,
Izv. Math. USSR, 37, no. 2, 1991, 243–272.
[4] A. V. Grishin, On the finite-basis property of abstract T -spaces, Fund. Prikl. Mat.,
1, 1995, 669–700 (Russian).
[5] W. A. Stein et al., Sage Mathematics Software (Version 4.6.1), The Sage Develop-
ment Team, 2011, http://www.sagemath.org.
Contact information
C. Bekh-Ochir Department of Mathematics, University of West-
ern Ontario, London, Ontario, Canada N6A 5B7
E-Mail: cbekhoch@gmail.com
S. Rankin Department of Mathematics, University of West-
ern Ontario, London, Ontario, Canada N6A 5B7
E-Mail: srankin@uwo.ca
URL: www.math.uwo.ca/members/
Received by the editors: 24.04.2012
and in final form 20.05.2012.
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