POLYNOMIAL IDENTITIES IN NIL-ALGEBRAS 1. Introduction Let F be

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 361, Number 11, November 2009, Pages 5629–5646 S 0002-9947(09)04977-0 Article electronically published on June 23, 2009

POLYNOMIAL IDENTITIES IN NIL-ALGEBRAS ELENA V. ALADOVA AND ALEXEI N. KRASILNIKOV

Abstract. We prove that in associative algebras over a field F of characteristic p ≥ 3 the polynomial identity x2p = 0 is not Specht. To prove this we construct a non-finitely based system of polynomial identities which contains the identity x2p = 0. We also give an example of a non-Specht polynomial identity of degree 2p in unital associative F -algebras.

1. Introduction Let F be a field, let A be a free associative algebra (without 1) over F on free generators x1 , x2 , . . . and let G be an associative F -algebra (with or without 1). Let f (x1 , . . . , xn ) ∈ A. We say that f (x1 , . . . , xn ) = 0 is a polynomial identity (or an identity) in G if f (g1 , . . . , gn ) = 0 for all g1 , . . . , gn ∈ G. Two systems of polynomial identities {ui = 0 | i ∈ I} and {vj = 0 | j ∈ J} are equivalent if every associative F -algebra satisfying all the identities ui = 0 satisfies all the identities vj = 0 and vice versa. If a system {ui = 0 | i ∈ I} is equivalent to some finite system of polynomial identities we say that the system {ui = 0 | i ∈ I} is finitely based or has a finite basis. We refer to [3], [6], [7], [13] and [16] for further terminology, basic facts and references concerning polynomial identities in associative algebras. A polynomial identity is called Specht if every system containing this identity has a finite basis. Note that over a field of characteristic 0 every system of polynomial identities is finitely based: this is a celebrated result of Kemer [14]. Therefore, over such a field, every polynomial identity is Specht. On the other hand, it has been proved by Belov [4], Grishin [9] and Shchigolev [17] that over a field of a prime characteristic p > 0 there are non-finitely based systems of polynomial identities and so there are identities which are not Specht. We study the following. Problem 1. For a given field F of characteristic p, find the smallest positive integer n = n(F ) such that the identity xn = 0 is not Specht. Note that over a field F of characteristic p, the identity xn = 0 with n < p is Specht. Indeed, according to the Nagata-Higman-Dubnov-Ivanov theorem ([15], [12], see also [8]) the identity xn = 0 (n < p) implies over F the identity of nilpotency x1 x2 . . . xk = 0 for some k = k(n) ∈ N and the latter identity is well known to be Specht (see, for instance, [2, Theorem 5.1.4]). Received by the editors June 7, 2006. 2000 Mathematics Subject Classification. Primary 16R10. The first author was partially supported by INTAS. The second author was partially supported by CNPq/FAPDF/PRONEX-Brazil, CNPq/ PADCT-Brazil, FINATEC-Brazil and RFBR-Russia. c 2009 American Mathematical Society

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ELENA V. ALADOVA AND ALEXEI N. KRASILNIKOV

On the other hand, the following polynomial identities in associative F -algebras have been proved to be non-Specht: (1) (2) (3) (4) (5) (6)

x32 = 0 over a field F of characteristic 2 (Grishin, 1999 [9], [10]); x6 = 0 over a field F of characteristic 2 (Gupta and Krasilnikov, 2002 [11]); 3 x2p (2p+1) = 0 over a field F of characteristic p ≥ 3 (Shchigolev, 1999 [17]); 3 2 x2p +p +1 = 0 over a field F of characteristic p ≥ 3 (Shchigolev, 2002 [19]); x12 = 0 over a field F of characteristic 3 (Aladova, 2002 [1]); x6p = 0 over a field F of characteristic p ≥ 5 (Aladova and Krasilnikov, 2003, unpublished).

Our main result is as follows. Theorem 1.1. Over a field F of characteristic p ≥ 3, the identity x2p = 0 is not Specht. We conjecture that n(F ) = 2p for each field F of a prime characteristic p ≥ 3; that is, the identity xn = 0 is Specht over F if n < 2p. Let p be a prime integer, p > 2. Let [x, y] = xy − yx, f (x, y) = xp−1 y p−1 [x, y], wn = wn (x1 , x2 , . . . , xn , y1 , y2 , . . . , yn )

p−1 . = [[x1 , x2 ], x3 ]f (x3 , y3 ) . . . f (xn , yn )[[y1 , y2 ], y3 ] [[x3 , x1 ], x2 ][[y3 , y1 ], y2 ]

We obtain Theorem 1.1 as a consequence of the following result. Theorem 1.2. Over a field F of characteristic p ≥ 3 the system of identities {wn = 0 | n = 3, 4 . . . } ∪ {x2p = 0} is not equivalent to any finite system of identities in associative F -algebras. To prove Theorem 1.2 we will construct, for each integer n ≥ 3, an associative F -algebra Bn such that Bn satisfies the identities x2p = 0 and wk = 0 for all k ≤ n but does not satisfy the identity wn+1 = 0. In the proof we use results of Shchigolev [18]. The algebras Bn and the identities wn = 0 in our paper are similar to, although different from, the corresponding algebras and the identities used by Belov [4]. We also use some ideas of [11]. We also study the following more general problem concerning non-Specht identities: Problem 2. For a given field F of characteristic p, find the lowest degree m = m(F ) of a non-Specht identity for associative F -algebras. One can consider Problem 2 for unital algebras (that is, for algebras with a unity 1) as well as for algebras which are not necessarily unital. Theorem 1.1 shows that for non-unital associative F -algebras we have m(F ) ≤ 2p. However, this theorem does not give any estimate for m(F ) for unital algebras. This is because a unital algebra does not satisfy any identity of the form xn = 0. Among the polynomial identities in unital associative F -algebras known to be non-Specht, the identity of the lowest degree is probably one that can be obtained from the proof of Belov’s result [4]. It has degree 5p + 2 (where p = char F ) and


POLYNOMIAL IDENTITIES IN NIL-ALGEBRAS

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is as follows: [x1 , x2 ][x3 , x4 , x5 ][x6 , x7 ][x8 , x9 , x10 ] . . . [x5p−2 , x5p−1 , x5p ][x5p+1 , x5p+2 ] = 0. So it was known for unital algebras that m(F ) ≤ 5p + 2. Let S2p be the permutation group on the set {1, 2, . . . , 2p} and let λ(x1 , x2 , . . . , x2p ) be the complete linearization of the polynomial x2p , λ(x1 , x2 , . . . , x2p ) = xσ(1) xσ(2) . . . xσ(2p) . σ∈S2p

We prove the following theorem. Theorem 1.3. Over a field F of characteristic p ≥ 3 the polynomial identity λ(x1 , x2 , . . . , x2p ) = 0 (of degree 2p) is not Specht in the class of unital associative F -algebras. Thus, for unital associative algebras we have m(F ) ≤ 2p as well as for non-unital ones. Theorem 1.3 is an immediate consequence of the following theorem. Theorem 1.4. Over a field F of characteristic p ≥ 3 the system of identities {wn = 0 | n = 3, 4, . . . } ∪ {λ(x1 , x2 , . . . , x2p ) = 0} is not equivalent to any finite system of polynomial identities in unital associative F -algebras.

Recall that

k l

=

k! l!(k−l)!

2. Auxiliary results if k ≥ l. Put kl = 0 if k < l. Let [x, y, z] = [[x, y], z].

Lemma 2.1. Let G be an associative ring satisfying the identity [x, y, z] = 0. Let t and k be integers such that t ≥ 1, k ≥ 0 and t + k > 1. Then, for any a1 , a2 , . . . , at−1 , r ∈ G, we have k r i1 a1 r i2 a2 r i3 . . . r it−1 at−1 r it = k+t−1 t−1 r a1 a2 . . . at−1 i1 +i2 +···+it =k i1 ,...,it ≥0

(2.1)

k−1 t−1 r + k+t−1 (t − j)a1 . . . aj−1 aj+1 . . . at−1 [aj , r] . t j=1

Proof. We will prove the equation (2.1) by induction on t and k. If t = 1, then (2.1) reduces to the equation r k = r k and if k = 0, then (2.1) reduces to a1 a2 . . . at−1 = a1 a2 . . . at−1 . Therefore, (2.1) holds for t = 1 and any k > 0 as well as for k = 0 and any t > 1. Let t > 1 and k > 0. Suppose that (2.1) holds for the pairs of integers (t, k − 1) and (t − 1, k). We are going to prove that it holds also for the pair (t, k). Let r i1 a1 r i2 a2 r i3 . . . r it−1 at−1 r it . σ= i1 +···+it =k i1 ,...,it ≥0


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It is clear that σ = σ1 + σ2 , where σ1 = r i1 a1 r i2 a2 r i3 . . . r it−1 at−1 r it · r, σ2 =

i1 +···+it =k−1 i1 ,...,it ≥0

r i1 a1 r i2 a2 r i3 . . . r it−2 at−2 r it−1 · at−1 .

i1 +···+it−1 =k i1 ,...,it−1 ≥0

Note that, for all a, b, r ∈ G, we have [a, r]b = b[a, r] and [a, r][b, r] = 0. Using these equations and the inductive hypothesis it is straightforward to check that k+t−2 k r a1 . . . at−1 σ1 = t−1 t−1 k + t − 2 k−1 + r a1 . . . aj−1 aj+1 . . . at−1 [aj , r] t−1 j=1

t−1 k + t − 2 k−1 + r (t − j)a1 . . . aj−1 aj+1 . . . at−1 [aj , r] t j=1

and

k+t−2 k σ2 = r a1 . . . at−1 t−2 t−2 k + t − 2 k−1 + r (t − j − 1)a1 . . . aj−1 aj+1 . . . at−1 [aj , r] . t−1 j=1

The result follows.

Lemma 2.2. Let F be a field of characteristic p ≥ 3 and let G be an associative F -algebra which satisfies the identities [x, y, z] = 0 and xp = 0. Let t and k be integers such that 1 ≤ t ≤ p − 1, k ≥ p − t + 1. Then, for all a1 , a2 , . . . , at−1 , r ∈ G, we have r i1 a1 r i2 a2 . . . r it−1 at−1 r it = 0. (2.2) i1 +i2 +···+it =k i1 ,...,it ≥0

Proof. Let t = 1. Then k ≥ p and the equation (2.2) reduces to the equation r k = 0, which holds since G satisfies the identity xp = 0. Let 2 ≤ t ≤ p − 1. Then, by Lemma 2.1, r i1 a1 r i2 a2 . . . r it−1 at−1 r it i1 +···+it =k i1 ,...,it ≥0

k+t−1 k k + t − 1 k−1 = r a1 a2 . . . at−1 + r ft , t−1 t

where ft ∈ G. If k ≥ p + 1, then the equation (2.2) holds because G satisfies the identity xp = 0 and so rk = rk−1 = 0. If p − t + 1 ≤ k ≤ p − 1, then k+t−1 (k+t−1)...(k+1) = = (k+t−1)...(k+1)k and k+t−1 are multiples of p. Since (t−1)! t! t−1 t char F = p, the equation (2.2) holds for such a k. Finally, if k = p, then r p = 0 p+t−1 = = (p+t−1)...(p+1)p is because G satisfies the identity xp = 0 and k+t−1 t! t t a multiple of p, so for k = p and 2 ≤ t ≤ p − 1 the equation (2.2) holds as well.



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Lemma 2.3. Let F be a field of characteristic p ≥ 3 and let k, l be integers such that k ≥ 1, 0 ≤ l ≤ p − 1. Let G be an associative F -algebra which satisfies the identities [x, y, z] = 0 and xp = 0. Then, for any a0 , a1 , r ∈ G, we have r i1 aj1 r i2 aj2 . . . r ip−1 ajp−1 r ip = 0. j1 +···+jp−1 =l i1 +i2 +···+ip =k i1 ,...,ip ≥0 0≤j1 ,...,jp−1 ≤1

Proof. Note that, for all k ≥ 1, k+p−1 k (2.3) r aj1 . . . ajp−1 = 0. p−1 Indeed, if 1 ≤ k ≤ p − 1, then k+p−1 p−1 a = 0 for each a ∈ G because char F = p. On the other hand, if k ≥ p, then r k = 0 because G satisfies the identity xp = 0. It follows from (2.3) and Lemma 2.1 that r i1 aj1 r i2 aj2 . . . r ip−1 ajp−1 r ip i1 +i2 +···+ip =k i1 ,...,ip ≥0

p−1 k + p − 1 k−1 = r (p − h)aj1 . . . ajh−1 ajh+1 . . . ajp−1 [ajh , r] . p h=1

Let σ=

r i1 aj1 r i2 aj2 . . . r ip−1 ajp−1 r ip .

j1 +···+jp−1 =l i1 +i2 +···+ip =k i1 ,...,ip ≥0 0≤j1 ,...,jp−1 ≤1

Then σ=

=

k + p − 1 k−1 r p

p−1

j1 +···+jp−1 =l 0≤j1 ,...,jp−1 ≤1

h=1

p−1 k + p − 1 k−1 r (p − h) p h=1

Let

aj1 . . . ajh−1 ajh+1 . . . ajp−1 [ajh , r] .

j1 +···+jp−1 =l 0≤j1 ,...,jp−1 ≤1

τ=

(p − h)aj1 . . . ajh−1 ajh+1 . . . ajp−1 [ajh , r]

aj1 . . . ajp−2 [ajp−1 , r].

j1 +···+jp−1 =l 0≤j1 ,...,jp−1 ≤1

It can be easily seen that τ=

aj1 . . . ajm−1 ajm+1 . . . ajp−1 [ajm , r]

j1 +···+jp−1 =l 0≤j1 ,...,jp−1 ≤1

for all m, 1 ≤ m ≤ p − 2. It follows that k + p − 1 k−1 p(p − 1) k + p − 1 k−1 τ. · (p−1)+(p−2)+· · ·+2+1 τ = r σ= r p p 2 Since char F = p, we have σ = 0.


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Lemma 2.4. Let F be a field of characteristic p ≥ 3 and let k be an integer, k ≥ 0. Let G be an associative F -algebra which satisfies the identity [x, y, z] = 0. Then, for any a, d0 , . . . , dk ∈ G, we have aj1 di1 aj2 . . . ajp−1 dip−1 ajp = 0. i1 +···+ip−1 =k j1 +···+jp =1 j1 ,...,jp ≥0 i1 ,...,ip−1 ≥0

Proof. Since char F = p, it follows from Lemma 2.1 that

j1

j2

a di1 a . . . a

jp−1

dip−1 a

jp

=

p−1

(p − s)di1 . . . dis−1 dis+1 . . . dip−1 [dis , a].

s=1

j1 +···+jp =1 j1 ,...,jp ≥0

Let σ=

aj1 di1 aj2 . . . ajp−1 dip−1 ajp .

i1 +···+ip−1 =k j1 +···+jp =1 j1 ,...,jp ≥0 i1 ,...,ip−1 ≥0

Then

p−1

i1 +···+ip−1 =k i1 ,...,ip−1 ≥0

s=1

σ=

(p − s)di1 . . . dis−1 dis+1 . . . dip−1 [dis , a]

=

p−1

(p − s)

s=1

Let τ=

di1 . . . dis−1 dis+1 . . . dip−1 [dis , a] .

i1 +···+ip−1 =k i1 ,...,ip−1 ≥0

di1 di2 . . . dip−2 [dip−1 , a].

i1 +···+ip−1 =k i1 ,...,ip−1 ≥0

It can be easily seen that τ= di1 . . . dip−3 dip−1 [dip−2 , a] = . . . i1 +···+ip−1 =k i1 ,...,ip−1 ≥0

··· =

di2 . . . dip−1 [di1 , a].

i1 +···+ip−1 =k i1 ,...,ip−1 ≥0

Therefore, σ = (p − 1)τ + · · · + (p − s)τ + · · · + τ =

p(p − 1) τ =0 2

since char F = p. 3. Construction of the algebra Bn

The lemma below follows immediately from a result of Shchigolev [18, Lemma xp−1 [x1 , x2 ]. 13]. Recall that f (x1 , x2 ) = xp−1 1 2 Lemma 3.1. Let F be a field of characteristic p ≥ 3. Then there exists a unital associative F -algebra R such that the following conditions are satisfied: 1. R as a vector space over F is a direct sum of its two-sided ideal I and the one-dimensional subspace generated by 1; 2. for each h ∈ I, we have hp = 0;



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3. R as an F -algebra with unity is generated by certain elements zi ∈ I (i ∈ N) such that, for each integer n > 0, the product f (z1 , z2 )f (z3 , z4 )f (z5 , z6 ) . . . f (z2n+1 , z2n+2 ) is not contained in the linear span of the set {1} ∪ {f (u1 , u2 ) . . . f (u2k−1 , u2k ) | 1 ≤ k ≤ n, u1 , . . . , u2k ∈ R};

(3.1)

4. [u, v, w] = 0 for all u, v, w ∈ R.

Notice that in [18] Shchigolev’s result was stated for an infinite field F . However, Shchigolev observed in [19] that the result remains valid for a finite field F as well: it is sufficient just to replace the expression “T -space” by “homogeneous T -space” in the proof. Let Mn be the F -linear span of the set (3.1). Define Rn to be the quotient algebra of the algebra of (2p + 1) × (2p + 1) matrices ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 0 0 ... 0 0 0

R R 0 0 0 ... 0 0 0

R R 0 0 0 ... 0 0 0

R R R R 0 ... 0 0 0

R R R R 0 ... 0 0 0

... ... ... ... ... ... ... ... ...

R R R R R ... 0 0 0

R R R R R ... R R 0

R R R R R ... R R 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

over the ideal ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 ... 0 0

0 0 0 ... 0 0

0 0 0 ... 0 0

... ... ... ... ... ...

0 0 0 ... 0 0

Mn 0 0 ... 0 0

⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠

that is, ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ Rn = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 0 0 ... 0 0 0

R R 0 0 0 ... 0 0 0

R R 0 0 0 ... 0 0 0

R R R R 0 ... 0 0 0

R R R R 0 ... 0 0 0

... ... ... ... ... ... ... ... ...

R R R R R ... 0 0 0

R R R R R ... R R 0


R/Mn R R R R ... R R 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

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Define Bn to be the subalgebra of Rn generated by the matrix ⎛ ⎜ ⎜ ⎜ ⎜ D=⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 ... 0 0 0

1 0 0 ... 0 0 0

0 1 0 ... 0 0 0

... ... ... ... ... ... ...

0 0 0 ... 0 0 0

0 0 0 ... 1 0 0

0 0 0 ... 0 1 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

and all the matrices ⎛

(3.2)

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ r=⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 0 0 ... 0 0 0

0 r 0 0 0 ... 0 0 0

0 0 0 0 0 ... 0 0 0

0 0 0 r 0 ... 0 0 0

0 0 0 0 0 ... 0 0 0

... ... ... ... ... ... ... ... ...

0 0 0 0 0 ... 0 0 0

0 0 0 0 0 ... 0 r 0

0 0 0 0 0 ... 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

where r ∈ I and I is the ideal of R mentioned in Lemma 3.1. We need the following observation. Remark 3.2. Let A, B and C be k × k, k × l and l × l matrices, respectively. Then (3.3)

A B 0 C

is a (k + l) × (k + l) matrix. Note that the mappings ψ

(1)

:

A 0

B C

→A

and

ψ

(2)

:

A B 0 C

→C

define homomorphisms of the algebra of the matrices of the form (3.3) onto the algebras of k × k and l × l matrices, respectively. It follows that the mapping ⎛

A11 ψ:⎝ 0 0

A12 A22 0

⎞ A13 A23 ⎠ → A22 A33

also defines a homomorphism of the corresponding associative algebra. Here A11 , A22 , A33 are k × k, l × l and m × m matrices, respectively, A12 and A23 are k × l and l × m matrices and A13 is a k × m matrix.



Lemma 3.3. Every element of the algebra ⎛ 0 u v ∗ ∗ ∗ ⎜ 0 r w h ∗ ∗ ⎜ ⎜ 0 0 0 u v ∗ ⎜ ⎜ 0 0 0 r w h ⎜ ⎜ 0 0 0 0 0 u ⎜ ⎜ 0 0 0 0 0 r ⎜ (3.4) ⎜ ... ... ... ... ... ... ⎜ ⎜ 0 0 0 0 0 0 ⎜ ⎜ 0 0 0 0 0 0 ⎜ ⎜ 0 0 0 0 0 0 ⎜ ⎝ 0 0 0 0 0 0 0 0 0 0 0 0

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Bn is a matrix of the form ∗ ∗ ∗ ∗ ∗ ∗ ... 0 0 0 0 0

... ... ... ... ... ... ... ... ... ... ... ...

∗ ∗ ∗ ∗ ∗ ∗ ... u r 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ ... v w 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ ... ∗ h u r 0

∗ ∗ ∗ ∗ ∗ ∗ ... ∗ ∗ v w 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

where u, v, w, h ∈ R and r ∈ I. Proof. Let X be an arbitrary ⎛ 0 a12 a13 ⎜ 0 a22 a23 ⎜ ⎜ 0 0 0 ⎜ ⎜ 0 0 0 ⎜ 0 0 X=⎜ ⎜ 0 ⎜ ... ... ... ⎜ ⎜ 0 0 0 ⎜ ⎝ 0 0 0 0 0 0

element of Rn . Let a14 a24 a34 a44 0 ... 0 0 0

a15 a25 a35 a45 0 ... 0 0 0

... ... ... ... ... ... ... ... ...

∗ ∗ ∗ ∗ ∗ ... 0 0 0

∗ ∗ ∗ ∗ ∗ ...

∗ ∗ ∗ ∗ ∗ ...

a2p−1,2p a2p,2p 0

a2p−1,2p+1 a2p,2p+1 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(entries denoted by ∗ are not important for the argument). Define mappings of the algebra Rn into the algebra of 5 × 5 matrices over R as follows: ⎛ ⎞ 0 a2k−1,2k a2k−1,2k+1 a2k−1,2k+2 a2k−1,2k+3 ⎜ 0 a2k,2k a2k,2k+1 a2k,2k+2 a2k,2k+3 ⎟ ⎜ ⎟ ⎜ 0 0 a2k+1,2k+2 a2k+2,2k+3 ⎟ ψk (X) = ⎜ 0 ⎟ (1 ≤ k ≤ p − 1). ⎝ 0 0 0 a2k+2,2k+2 a2k+2,2k+3 ⎠ 0 0 0 0 0 By Remark 3.2, ψ1 , ψ2 , . . . , ψp−1 are homomorphisms of Note that ⎛ 0 1 ⎜ 0 0 ⎜ ψ1 (D) = ψ2 (D) = · · · = ψp−1 (D) = ⎜ ⎜ 0 0 ⎝ 0 0 0 0 and

⎛ ⎜ ⎜ ψ1 (r) = ψ2 (r) = · · · = ψp−1 (r) = ⎜ ⎜ ⎝

0 0 0 0 0

0 r 0 0 0

the F -algebra Rn .

0 0 0 0 0


0 1 0 0 0

0 0 1 0 0 0 0 0 r 0

0 0 0 1 0 0 0 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎟ ⎠

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for all matrices r (r ∈ I) of the form (3.2). Since the algebra Bn is generated by D and the matrices of the form (3.2), we have ψ1 (X) = ψ2 (X) = · · · = ψp−1 (X) for each X ∈ Bn . Comparing the matrices ψ1 (X) = ψ2 (X) = · · · = ψp−1 (X), we have a22 = a44 = · · · = a(2p−2)(2p−2) and also a44 = a66 = · · · = a2p,2p , so a22 = a44 = · · · = a2p,2p . Similarly, a12 = a34 = · · · = a(2p−1)2p , a23 = a45 = · · · = a2p(2p+1) , a13 = a35 = · · · = a(2p−1)(2p+1) and a24 = a46 = · · · = a(2p−2)2p . 4. Proof of Theorem 1.2 First we are going to check that the algebra Bn satisfies the identities wk = 0 for all k ≤ n but does not satisfy the identity wn+1 = 0. (j) Let Xi (i = 1, 2, . . . , k, j = 1, 2) be arbitrary elements of Bn . Let ⎛ ⎞ (j) 0 ui ∗ ∗ ∗ ... ∗ ∗ ∗ ⎜ ⎟ ⎜ 0 ri(j) vi(j) ∗ ∗ ... ∗ ∗ ∗ ⎟ ⎜ ⎟ (j) ⎜ 0 ∗ ... ∗ ∗ ∗ ⎟ 0 0 ui ⎜ ⎟ (j) (j) ⎜ 0 ⎟ 0 0 r v . . . ∗ ∗ ∗ ⎜ ⎟ i i (j) (4.1) Xi = ⎜ 0 0 0 0 ... ∗ ∗ ∗ ⎟ ⎜ 0 ⎟, ⎜ ... ... ... ... ... ... ... ... ... ⎟ ⎜ ⎟ ⎜ ⎟ (j) ⎜ 0 0 0 0 0 ... 0 ui ∗ ⎟ ⎜ (j) (j) ⎟ ⎝ 0 vi ⎠ 0 0 0 0 ... 0 ri 0 0 0 0 0 ... 0 0 0 (j)

(j)

(j)

∈ R. Then ⎛ 0 u ¯(j) ⎜ 0 0 ⎜ ⎜ 0 0 ⎜ ⎜ 0 0 (j) (j) (j) [X1 , X2 , X3 ] = ⎜ ⎜ ... ... ⎜ ⎜ 0 0 ⎜ ⎝ 0 0 0 0 ⎛ 0 u ˜(j) ⎜ 0 0 ⎜ ⎜ 0 0 ⎜ ⎜ 0 0 (j) (j) (j) [X3 , X1 , X2 ] = ⎜ ⎜ ... ... ⎜ ⎜ 0 0 ⎜ ⎝ 0 0 0 0

where ri

∈ I, ui , vi

∗ v¯(j) 0 0 ... 0 0 0

∗ ∗ u ¯(j) 0 ... 0 0 0

... ... ... ... ... ... ... ...

∗ ∗ ∗ ∗ ... 0 0 0

∗ ∗ ∗ ∗ ... u ¯(j) 0 0

∗ ∗ ∗ ∗ ... ∗ (j) v¯ 0

∗ v˜(j) 0 0 ... 0 0 0

∗ ∗ u ˜(j) 0 ... 0 0 0

... ... ... ... ... ... ... ...

∗ ∗ ∗ ∗ ... 0 0 0

∗ ∗ ∗ ∗ ... u ˜(j) 0 0

∗ ∗ ∗ ∗ ... ∗ v˜(j) 0


⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎠


5639

where (j) (j)

(j) (j)

(j)

(j)

(j)

(j)

u ¯(j) = (u1 r2 − u2 r1 )r3 − u3 [r1 , r2 ], (j)

(j)

(j)

(j)

(j) (j)

(j) (j)

v¯(j) = [r1 , r2 ]v3 − r3 (r1 v2 − r2 v1 ), (4.2) (j) (j)

(j) (j)

(j)

(j)

(j)

(j)

u ˜(j) = (u3 r1 − u1 r3 )r2 − u2 [r3 , r1 ], (j)

(j)

(j)

(j)

(j) (j)

(j) (j)

v˜(j) = [r3 , r1 ]v2 − r2 (r3 v1 − r1 v3 ). (j)

(j)

(j)

(j)

(j)

(j)

The matrices [X1 , X2 , X3 ] and [X3 , X1 , X2 ] are nil-triangular since the algebra R satisfies the identity [x, y, z] = 0. (1) (1) (1) (2) (2) (2) Let H = [X3 , X1 , X2 ][X3 , X1 , X2 ]. Then ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ H =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛

H p−1

⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎝

0 0 0 0 0 ... 0 0 0

0 0 0 0 0 ... 0 0 0

u ˜(1) v˜(2) 0 0 0 0 ... 0 0 0

0 0 0 0 ... 0

··· ··· ··· ··· ... ···

0 0 0 0 ... 0

∗ (1) (2) ˜ v˜ u 0 0 0 ... 0 0 0

(˜ u(1) v˜(2) )p−1 0 0 0 ... 0

∗ ∗ u ˜(1) v˜(2) 0 0 ... 0 0 0

∗ ∗ ∗ ∗ ∗ ... 0 0 0

... ... ... ... ... ... ... ... ...

∗ ˜(2) )p−1 (˜ v (1) u 0 0 ... 0

∗ ∗ ∗ ∗ ∗ ... 0 0 0

∗ ∗ ∗ ∗ ∗ ... u ˜(1) v˜(2) 0 0 ⎞

∗ ∗ (˜ u(1) v˜(2) )p−1 0 ... 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

and ⎛ ⎜

⎜ (2) (2) (2) [X1 , X2 , X3 ]H p−1 = ⎜ ⎜ ⎝

0 0 0 ... 0

··· ··· ··· ... ···

0 0 0 ... 0

v (1) u ˜(2) )p−1 u ¯(2) (˜ 0 0 ... 0

∗ u(1) v˜(2) )p−1 v¯(2) (˜ 0 ... 0

Further, ⎛ ⎜ ⎜ ⎜ ⎜ (1) (2) (1) (2) f (X3 , X3 ) . . . f (Xk , Xk ) = ⎜ ⎜ ⎜ ⎜ ⎝

0 0 0 ... 0 0 0

∗ r˜ 0 ... 0 0 0

∗ ∗ 0 ... 0 0 0

... ... ... ... ... ... ...


∗ ∗ ∗ ... 0 0 0

∗ ∗ ∗ ... ∗ r˜ 0

∗ ∗ ∗ ... ∗ ∗ 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

5640

ELENA V. ALADOVA AND ALEXEI N. KRASILNIKOV (1)

(2)

(1)

(2)

where r˜ = f (r3 , r3 ) . . . f (rk , rk ). It follows that

⎛

⎜ ⎜ (1) (2) (1) (2) (2) (2) (2) f (X3 , X3 ) . . . f (Xk , Xk )[X1 , X2 , X3 ]H p−1 = ⎜ ⎜ ⎝ (1)

(2)

(1)

··· ··· ··· ... ···

0 0 0 ... 0

0 0 0 ... 0

∗ y 0 ... 0

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

(2)

where y = f (r3 , r3 ) · · · f (rk , rk )¯ v (2) (˜ u(1) v˜(2) )p−1 . Therefore, if (1)

(1)

(1)

(2)

(1)

(2)

(2)

(2)

W = wk (X1 , X2 , . . . , Xk , X1 , X2 , . . . , Xk ) (1)

(1)

(1)

(1)

(2)

(2)

(2)

(2)

= [X1 , X2 , X3 ]f (X3 , X3 ) · · · f (Xk , Xk )[X1 , X2 , X3 ] (p−1) (1) (1) (1) (2) (2) (2) , × [X3 , X1 , X2 ][X3 , X1 , X2 ] then

⎞ d + Mn ⎟ 0 ⎟, (4.3) ⎠ ... 0 p−1 (1) (2) (1) (2) ˜(1) v˜(2) ¯(1) f (r3 , r3 ) . . . f (rk , rk )¯ v (2) u . By (4.2), we have with d = u ¯(1) y = u (1) (1) (1) (1) (1) (1) (1) (1) (1) (2) (1) (2) d = (u1 r2 − u2 r1 )r3 − u3 [r1 , r2 ] f (r3 , r3 ) . . . f (rk , rk ) (2) (2) (2) (2) (2) (2) (2) (2) · [r1 , r2 ]v3 − r3 (r1 v2 − r2 v1 ) p−1 (1) (1) (1) (1) (1) (1) (1) (1) (2) (2) (2) (2) (2) (2) (2) (2) . · (u3 r1 −u1 r3 )r2 −u2 [r3 , r1 ] [r3 , r1 ]v2 −r2 (r3 v1 −r1 v3 ) ⎛

0 ⎜ 0 W =⎜ ⎝ ... 0

0 0 ... 0

... ... ... ...

0 0 ... 0

Since the identity [x, y, z] = 0 implies [x, z][y, z] = 0, the element f (ri , rj ) belongs to the centre of R for all ri , rj ∈ R. Also, ri ∈ I, so, by Lemma 3.1, (ri )p = 0. Therefore, (1)

(1)

(1)

(1)

(2)

(1)

(2)

(2)

(2)

(2)

d = −u3 [r1 , r2 ]f (r3 , r3 ) . . . f (rk , rk ) · [r1 , r2 ]v3 p−1 (1) (1) (1) (1) (1) (2) (2) (2) (2) (2) · (u3 r1 − u1 r3 )r2 r2 (r3 v1 − r1 v3 ) (1) (2) (1) p−1 (1) (1) (2) p−1 (2) (2) (1) (2) (1) (2) = −u3 v3 r2 [r1 , r2 ] r2 [r1 , r2 ]f (r3 , r3 ) . . . f (rk , rk ) p−1 (1) (1) (1) (1) (2) (2) (2) (2) · (u3 r1 − u1 r3 )(r3 v1 − r1 v3 ) (1) (2) (1) p−1 (1) (1) (2) p−1 (2) (2) = −u3 v3 r2 [r1 , r2 ] r2 [r1 , r2 ] p−1 (1) (2) (1) (2) (1) (1) (2) (2) · f (r3 , r3 ) . . . f (rk , rk ) u3 r1 r1 v3 (1) p−1 (1) p−1 (1) (1) (2) p−1 (2) p−1 (2) (2) r2 r2 = − r1 [r1 , r2 ] r1 [r1 , r2 ] p (1) (2) (1) (2) (1) (2) · f (r3 , r3 ) . . . f (rk , rk ) u3 v3 . Since R is a direct sum of the vector spaces I and 1F , for each r ∈ R we have r = α + ρ, where α ∈ F , ρ ∈ I. It follows that r p = (α + ρ)p = αp ∈ F (recall that ρp = 0 for each ρ ∈ I). Then (1)

(1)

(2)

(2)

(1)

(2)

(1)

(2)

d = −αp f (r1 , r2 )f (r1 , r2 )f (r3 , r3 ) . . . f (rk , rk ).



5641

If k ≤ n, then, by the definition of Mn , we have (1)

(1)

(2)

(2)

(1)

(2)

(1)

(2)

f (r1 , r2 )f (r1 , r2 )f (r3 , r3 ) . . . f (rk , rk ) ∈ Mn , so d ∈ Mn and W = 0, that is, (1)

(1)

(1)

(2)

(2)

(2)

wk (X1 , X2 , . . . , Xk , X1 , X2 , . . . , Xk ) = 0 (j)

for all Xi ∈ Bn (i = 1, 2, . . . , k, j = 1, 2). Thus, the algebra Bn satisfies the identities wk = 0 for all k ≤ n. In order to prove that Bn does not satisfy the identity wn+1 = 0, consider the element W = wn+1 (z1 , z2 , D + z5 , z7 , . . . , z2n+1 , z3 , z4 , D + z6 , z8 , . . . , z2n+2 ) = [z1 , z2 , D + z5 ]f (D + z5 , D + z6 )f (z7 , z8 ) . . . f (z2n+1 , z2n+2 ) (p−1) × [z3 , z4 , D + z6 ] [D + z5 , z1 , z2 ][D + z6 , z3 , z4 ] , where, for each i, zi is the matrix of the form (3.2) corresponding to the generator zi of R mentioned in Lemma 3.1. By the calculations above, W is of the form (4.3) with d = −f (z1 , z2 )f (z3 , z4 )f (z5 , z6 ) . . . f (z2n+1 , z2n+2 ). By Lemma 3.1, d ∈ / Mn . Thus, W = 0, so the algebra Bn does not satisfy the identity wn+1 = 0. Now to complete the proof of Theorem 1.2 it remains to check that Bn satisfies the identity x2p = 0, where p = char F . Let Eij be the matrix units (1 ≤ i, j ≤ 2p + 1); that is, Eij has 1 at the position (i, j) and 0 elsewhere. It is clear that 0, if j = k; Eij Ekl = Eil , if j = k. For each X ∈ Bn ,

X=

aij Eij ,

1≤i≤j≤2p+1

where aii ∈ I (i = 2, 4, . . . , 2p), aii = 0 (i = 1, 3, 5, . . . , 2p + 1), aij ∈ R if i < j and (i, j) = (1, 2p + 1) and a1,2p+1 ∈ R/Mn . Since I satisfies the identity xp = 0, the matrix X 2p is nil-triangular. It can be written as follows: m i j E i j , X 2p = 1≤i p − t = 0. By Lemma 2.3, if k ≥ 1 and p − 1 ≤ l ≤ 2p − 2, then every sum of the form (4.10) is equal to 0. Therefore, mi j = 0 if either i = 1 or j = 2p + 1. Now consider m1,2p+1 . It is clear that m1,2p+1 = d1 + d2 + Mn , where d1 is the sum of all the elements of the form (4.9) such that deg (b0 b1 . . . bt ) < p + t and d2 is the sum of all such elements with deg (b0 b1 . . . bt ) = p + t. Using the argument above it is easy to prove that d1 = 0. Hence, m1,2p+1 = d2 + Mn . For every element of the form (4.9) with deg (b0 b1 . . . bt ) = p + t, the corresponding sequence (4.7) contains all the odd numbers from 1 to 2p + 1. Since, by Lemma 3.3, a2i−1,2i = a12 , a2i−1,2i+1 = a13 , a2i,2i+1 = a23 for all i, 1 ≤ i ≤ p, the terms bj in the sum (4.9) are of the form b0 = as(0) s(0) . . . as(0) (0) c s

= ac130 −1 a12 ,

bj = as(j) s(j) . . . asc(j) s(j)

= a23 a13j

bt = as(t) s(t) . . . asc(t) s(t)

ct −1 = a23 a13 .

1

1

1

2

0

2

j

2

t

c0 +1

cj +1

ct +1

c −2

a12

(1 ≤ j ≤ t − 1),

It follows that the sum (4.9) reduces to ct −1 (4.11) ac130 −1 a12 r i1 a23 ac131 −2 a12 . . . r it a23 a13 , i1 +···+it =k i1 ,...,it ≥0

where k = 2p − (p + t) = p − t, 1 ≤ t ≤ p. Since c0 + · · · + ct = p + t, we have (4.12) (c0 − 1) + (c1 − 2) + · · · + (ct−1 − 2) + (ct − 1) = p + t − 2 − 2(t − 1) = p − t. Let σt0 be the sum of all the elements (4.11) of d2 with t = t0 . Then σt + Mn m1,2p+1 = d2 + Mn = 1≤t≤p

and, by (4.12), σt =

j0 +···+jt =p−t i1 +···+it =p−t j0 ,...,jt ≥0 i1 ,...,it ≥0

=

aj130 (a12 r i1 a23 )aj131 . . . (a12 r it a23 )aj13t

aj130 (a12 r i1 a23 )aj131 . . . (a12 r it a23 )aj13t ,

i1 +···+it =p−t j0 +···+jt =p−t i1 ,...,it ≥0 j0 ,...,jt ≥0

where 1 ≤ t ≤ p. If 1 ≤ t ≤ p − 2, then it follows from Lemma 2.1 and the equation char F = p that aj130 (a12 r i1 a23 )aj131 . . . (a12 r it a23 )aj13t = 0. j0 +···+jt =p−t j0 ,...,jt ≥0


5644


(Note that to apply Lemma 2.1 to the sum above it is convenient to replace j0 , . . . , jt with j1 , . . . , jt+1 .) Hence, we have σ1 = · · · = σp−2 = 0. By Lemma 2.4, σp−1 = 0. Let t = p. Then k = p − t = 0, so p σp = a12 a23 a12 a23 . . . a23 = a12 a23 . Therefore,

p m1,2p+1 = a12 a23 + Mn .

By Lemma 3.1, for each w ∈ R there are α ∈ F and r ∈ I such that w = α(1+r). Then wp = αp (1 + r p ) = αp since r p = 0. Hence, by the definition of Mn , we have wp ∈ Mn for any w ∈ R and any n ≥ 1. In particular, (a12 a23 )p ∈ Mn , so, for any n ≥ 1, we have m1,2p+1 = Mn , that is, m1,2p+1 = 0 ∈ R/Mn . It follows that X 2p = 0, where X is an arbitrary element of Bn . Thus, for each n ≥ 1, the algebra Bn satisfies the identity x2p = 0. This completes the proof of Theorem 1.2. 5. Proof of Theorem 1.4 Let Bn+ be the F -algebra obtained from Bn by formally adjoining a unity element 1. Then as a vector space over F the algebra Bn+ is a direct sum of (its two-sided ideal) Bn and the one-dimensional subspace generated by 1. We will prove Theorem 1.4 by checking that, for each n ≥ 3, the unital algebra Bn+ satisfies the identities λ = 0 and wk = 0 for k ≤ n but does not satisfy the identity wn+1 = 0. First we check that Bn+ satisfies the identity λ = 0. We need the following two observations. 1. Since Bn satisfies the polynomial identity x2p = 0 and λ = 0 is the complete linearization of x2p = 0, the algebra Bn satisfies the identity λ = 0. 2. Since F is a field of characteristic p, we have xτ (1) . . . xτ (2p−1) = 0. λ(x1 , . . . , x2p−1 , 1) = 2p τ ∈S2p−1

Similarly, for each i, λ(x1 , . . . , xi−1 , 1, xi+1 , . . . , x2p ) = 0. Now let Yi (i = 1, . . . , 2p) be arbitrary elements of Bn+ . Then Yi = αi + Xi , where αi ∈ F and Xi ∈ Bn for all i. The polynomial λ is multilinear, so, by the observations above, λ(Y1 , . . . , Y2p ) = λ(α1 + X1 , . . . , α2p + X2p ) = λ(X1 , . . . , X2p ) = 0. Thus, the algebra Bn+ satisfies the identity λ = 0, as required. Now we need to check that, for each n ≥ 3, the algebra Bn+ satisfies wk = 0 for k ≤ n but does not satisfy wn+1 = 0. Since the identity wn+1 = 0 is not satisfied in Bn , it obviously is not satisfied in Bn+ . Therefore, to complete the proof of Theorem 1.4 it suffices to prove that Bn+ satisfies the identity wk = 0 for k ≤ n. (j) (j) Let Yi (i = 1, 2, . . . , k, j = 1, 2) be arbitrary elements of Bn+ . Then Yi = (j) (j) (j) (j) αi + Xi , where αi ∈ F and Xi ∈ Bn for all i, j. Let (1)

(1)

(1)

(2)

(2)

(2)

W = wk (Y1 , Y2 , . . . , Yk , Y1 , Y2 , . . . , Yk ).



5645

Then (1)

(1)

(1)

(1)

(2)

(1)

(2)

(2)

(2)

(2)

W = [Y1 , Y2 , Y3 ]f (Y3 , Y3 ) · · · f (Yk , Yk )[Y1 , Y2 , Y3 ] (p−1) (1) (1) (1) (2) (2) (2) × [Y3 , Y1 , Y2 ][Y3 , Y1 , Y2 ] (1)

(1)

(1)

(1)

(1)

(2)

(2)

(1)

(1)

(2)

(2)

= [X1 , X2 , X3 ]f (α3 + X3 , α3 + X3 ) · · · f (αk + Xk , αk + Xk ) (p−1) (2) (2) (2) (1) (1) (1) (2) (2) (2) . × [X1 , X2 , X3 ] [X3 , X1 , X2 ][X3 , X1 , X2 ] (j)

Suppose that Xi are as in (4.1). It follows from the calculations in the proof of Theorem 1.2 that W is of the form (4.3) with (1)

(1)

(2)

(2)

(1)

(1)

(2)

(2)

(1)

(1)

(2)

(2)

d = −αp f (r1 , r2 )f (r1 , r2 )f (α3 + r3 , α3 + r3 ) . . . f (αk + rk , αk + rk ). If k ≤ n, then, by the definition of Mn , we have (1)

(1)

(2)

(2)

(1)

(2)

(1)

(2)

f (s1 , s2 )f (s1 , s2 )f (s3 , s3 ) . . . f (sk , sk ) ∈ Mn (j)

for all si

∈ R. Therefore, d ∈ Mn and W = 0, that is, (1)

(1)

(1)

(2)

(2)

(2)

wk (Y1 , Y2 , . . . , Yk , Y1 , Y2 , . . . , Yk ) = 0 (j)

for all Yi ∈ Bn+ (i = 1, 2, . . . , k, j = 1, 2). Thus, the algebra Bn+ satisfies the identities wk = 0 for all k ≤ n, as required. This completes the proof of Theorem 1.4. Acknowledgement Thanks are due to the referee whose valuable remarks and suggestions improved the paper considerably. References 1. E. V. Aladova, A non-finitely based variety of nil-algebras over a field of characteristic 3. (Russian) Algebra and linear optimization (Ekaterinburg, 2002), Ross. Akad. Nauk Ural. Otdel., Inst. Mat. Mekh., Ekaterinburg, 2002, pp. 5–11. MR1847556 (2003e:16026) 2. Yu. A. Bahturin, Identical relations in Lie algebras, VNU Science Press, Utrecht, 1987. MR886063 (88f:17032) 3. Yu. A. Bakhturin, A. Yu. Ol’shanskii, Identities, Algebra, II. Encyclopaedia of Mathematical Sciences, Current Problems in Mathematics, Fundamental Directions, vol. 18, Springer-Verlag, Berlin, 1991, pp. 117–240. MR1121267 (92b:00057) 4. A. Ya. Belov, On non-Specht varieties (Russian), Fundam. Prikl. Mat. 5 (1999), 47–66. MR1799544 (2001k:16040) 5. A. Ya. Belov, Counterexamples to the Specht problem (Russian). Mat. Sb. 191 (2000), 13–24. English translation in Sb. Math. 191 (2000), 329–340. MR1773251 (2001g:16043) 6. L. A. Bokyt’, I. V. L’vov, V. K. Harchenko, Noncommutative rings, Algebra, II. Encyclopaedia of Mathematical Sciences, Current Problems in Mathematics, Fundamental Directions, vol. 18, Springer-Verlag, Berlin, 1991, pp. 5–116. MR1121267 (92b:00057) 7. V. Drensky, Free algebras and PI-algebras. Springer-Verlag Singapore, Singapore, 2000. MR1712064 (2000j:16002) 8. E. Formanek, The Nagata-Higman theorem, Acta Appl. Math. 21 (1990), 185–192. MR1085778 (92d:15023) 9. A. V. Grishin, Examples of T -spaces and T -ideals of characteristic 2 without the finite basis property (Russian). Fundam. Prikl. Mat. 5 (1999), 101–118. MR1799541 (2002a:16028) 10. A. V. Grishin, On non-Spechtianness of the variety of associative rings that satisfy the identity x32 = 0, Electron. Res. Announc. Amer. Math. Soc. 6 (2000), 50–51 (electronic). MR1777855


5646


11. C. K. Gupta, A. N. Krasilnikov, A non-finitely based system of polynomial identities which contains the identity x6 = 0, Quart. J. Math. 53 (2002), 173–183. MR1909509 (2003c:16031) 12. G. Higman, On a conjecture of Nagata, Proc. Cambridge Phil. Soc. 52 (1956), 1–4. MR0073581 (17:453e) 13. A. Kanel-Belov, L. H. Rowen, Computational aspects of polynomial identities, A K Peters, Ltd., Wellesley, MA, 2005. MR2124127 (2006b:16001) 14. A. R. Kemer, Finite basability of identities of associative algebras (Russian), Algebra i Logika 26 (1987), 597–641. MR985840 (90b:08008) 15. M. Nagata, On the nilpotency of nil algebras. J. Math. Soc. Japan 4 (1953), 296–301. MR0053088 (14:719g) 16. L. H. Rowen, Ring theory, Academic Press, Inc., Boston, MA, 1991. MR1095047 (94e:16001) 17. V. V. Shchigolev, Examples of infinitely based T -ideals (Russian), Fundam. Prikl. Mat. 5 (1999), 307–312. MR1799533 (2001k:16044) 18. V. V. Shchigolev, Examples of infinitely basable T -spaces (Russian), Mat. Sb. 191 (2000), 143–160. English translation in Sb. Math. 191 (2000), 459–476. MR1773258 (2001f:16044) 19. V. V. Shchigolev, Infinitely based T -spaces and T -ideals, Ph.D. thesis, Moscow State University, 2002. Department of Algebra, Moscow Pedagogical State University, 14 Krasnoprudnaya St., Moscow 107140, Russia Department of Mathematics, University of Bras´ılia, 70910-900, Bras´ılia-DF, Brazil E-mail address: [email protected]


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