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Chapter 16 Right Alternative Algebras
CHAPTER
16
Right Alternative Algebras
Along with alternative algebras, there are often encountered in mathematics algebras with weaker conditions of associativity.The most important of these conditions is power-associativity,that is, every subalgebra generated by a single element is associative. The study of this broad class of algebras (in particular, it contains all anticommutative algebras) meets with insurmountable obstacles. However, some of its subclasses, which also contain the alternative algebras, have been adequately studied. Right alternative algebras, which arose from the study of a certain class of projective planes, have been studied the most intensively. In this chapter we shall briefly set forth the basic facts which are known up to the present time about right alternative algebras.
1. ALGEBRAS WITHOUT NILPOTENT ELEMENTS
An algebra is called right alternative if it satisfies the identities
34"
16.1.
343
ALGEBRAS WITHOUT NILPOTENT ELEMENTS
We recall that (1) is called the right alternative identity, and (2) is the right Moufang identity. By Proposition 1.4 the linearizations of these identities
+ ( X 4 Y = X ( Y " 4, ( ( X Y ) Z ) t + ( ( X W Y = x ( ( y z ) t + (tzly),
(1')
(XY)Z
(2')
are also valid in every right alternative algebra. If the algebra is without elements of additive order 2, then the right Moufang identity is a consequence of the right alternative identity. In fact, we have 2(yz)y = (z y ) y - z y2, whence by (1') and (1) 0
0
0
Y l - x(z Y 2 ) = [.(Z Y)lY + ( X Y ) ( Z Y ) - ( X Z ) Y 2 - ( X Y 2 ) Z = C(XZ)YIY + [(XY)ZIY + [ ( X Y ) Z l Y + C(XY)YlZ
2XC(YZ)Yl = .[(Z
O
Y)
O
O
O
- ( X Z ) Y 2 - (XY%
= 2[(XY)ZlY.
We note that the right Moufang identity is equivalent to the identity ( x , Y Z , Y ) = (x, z, Y)Y.
(3)
Actually, we have
0 = [ ( X Y ) Z I Y - X [ ( Y Z ) Y ] = ( x , Y, Z)Y
+ ( x , YZ7 Y )
= ( x , Y Z , Y ) - (x, z, Y)Y.
THEOREM 1. Every right alternative algebra is power-associative. PROOF. It suffices to prove that for arbitrary nonassociative words
ui = ui(x), where i = 1,2, 3, and an arbitrary element a of a right alternative algebra A, it is true (fi,,ii27fi3)= 0 where tii = ui(a).We shall carry out the
proof by induction on the total length d(u,) in any algebra there is the equality ( X Y , 2, t )
- (x, Y Z , r ) + (x, Y,
+ d(u2)+ d(u3). By the fact that
= X ( Y , z, t )
+ (x, Y ,z ) ~ ,
we can assume that u3(x)= x. Furthermore, by the induction assumption one can assume that u2 = x . u ; . It is now clear that in view of (3), (El,E2,fi3)
= @,,a
.ii;,u)
= 0.
This proves the theorem. With a view toward a compact presentation of calculations, we shall make use of notation which was introduced into use by Smiley. We denote by c' the operator of right multiplication R,. We then set cd = (cd)' - c'd' and
344
16.
RIGHT ALTERNATIVE ALGEBR /is
cd = (cd)’ - d’c’. Identities (I), (l’),(2), and (2’) imply the relations
+
(cd + dc)’ = c’d’ d’c‘, c‘ = 0, cd d‘ = 0, c’d‘c‘ = [ ( c d ) ~ ] ’ , c‘d‘s’ s’d’c’ = [ ( c d ) ~ ( s d ) ~ ] ’ . c’c‘ = (c2)1,
+
+
+
THEOREM 2 (Mikheeu). In every right alternative algebra the (Mikheev) identity (x, x, YI4 = 0
is valid. PROOF. Let a, b, c be arbitrary elements of a right alternative algebra A. We set q = [a, b] and p = (a,a, b). We shall prove the relations
abab =
aba‘ab =
- (4,a, b)’, - (qa, a, b)’,
Relation (4) is obvious. Moreover,
+
(ab)’(ab)’- (ab)‘b’a’- a’b’(ab)’ a’b’ba‘ = [(ab)(ab)]’- [((ab)b)a (ab)(ab)]‘ [(ab2)a]’= 0.
abab =
+
+
The second part of ( 5 ) is linearization of the first. We shall now prove (’7):
+ +
aba‘ab = (ab)’d(ab)‘- (ab)’a’a’b’- b’a’a’(ab)’ ba’a‘a‘b‘ = ([(ab)a](ab) - [(ab)a2]b- (ba2)(ab) (ba3)b}‘ = - (qa,a, b)’.
Relation (6) is proved analogously. Now by (4),(5), (6), and (7) p(4, a, b) =
a, b)’ = aababab= 0,
- aab(q,
P(qa, a, b) =
a, b)’ = aababdab= 0.
In addition, we have the relations p‘ = abo- dab = abo - aha'.
We shall only prove the first: p‘ = -(a, b, a)’ = [a(ba)- (&)a]’ = [ ~ ( b a ) ]-‘ a‘b’a‘ = [ ~ ( b a ) ] ’ da’b‘ - a’(ba)’ - a’(ab)’ = abU- dab.
+
(8) (9)
16.1.
ALGEBRAS WITHOUT NILPOTENT ELEMENTS
345
COROLLARY 1 (Kleinfeld). Every right alternative algebra without nilpotent elements is alternative.
A much stronger assertion, due to Albert [3, 71, is valid for finitedimensional algebras. He proved that every finite-dimensional right alternative algebra over a field of characteristic 2 2 , which does not contain nil-ideals, is alternative. The proof is technically complicated, and we shall not produce it here. COROLLARY 2 (Skornyakou). Every right alternative division algebra is alternative. Skornyakov proved by means of this result that the realization of little Desargues’ theorem on two lines of a projective plane implies its projective realization in that plane. The question on the structure of simple right alternative algebras is at present not clear to its conclusion. Thedy proved that under certain additional restrictions, simple right alternative algebras are alternative [232, 2331. For example, this is so if the simple algebra A contains an idempotent e # 0, 1 for which (e,e, A ) = (0).There was the conjecture that simple right alternative algebras which are not alternative do not exist. However, Mikheev [152] recently disproved this conjecture by constructing an appropriate example. It is interesting that the algebra in Mikheev’s example satisfies the identity x3 = 0, that is, is a nil-algebra of bounded index. The hope, nevertheless, remains that right alternative algebras which do not contain nil-ideals are alternative. Some basis for such a conjecture is the above-mentioned result of Albert. We now study in somewhat more detail alternative(and consequently, also right alternative) algebras without nilpotent elements. LEMMA 1. In every alternative algebra without nilpotent elements
(ab)c = 0 0 a(bc) = 0.
(1 1)
16.
346
RIGHT ALTERNATIVE ALGEBRAS
PROOF. If (ab)c = 0, then using the middle Moufang identity we have
[ ~ ( b c )=] a(bc ~ - (U . bc . a) * bc) = a(bc . (ab * CU) * bc) = a((bc * ab)(ca . bc)) = a((bc . ab)(c . ab . c ) ) = 0, whence a(bc) = 0. The reverse implication is proved analogously. Algebras satisfying condition (1 1) are called associatiue modulo zero. They were introduced into consideration by Ryabukhin. It is easy to see that if in an associative modulo zero algebra a product of some n elements equals zero for one arrangement of parentheses, then it also equals zero for any other arrangement. LEMMA 2. In every associative modulo zero algebra without nilpotent elements ab = 0
ba = 0.
(12)
PROOF. ab = 0 * b(ab)= 0 * (ba)b= 0 => [(ba)b]a= 0 ba = 0.
(ba)2= 0
LEMMA 3. Let u = u(xl, . . . ,x,) be a multilinear nonassociative word. If u(al,u2, . . . , a,) = 0 for elements a,, a 2 , . . . , a, of an associative modulo zero algebra A without nilpotent elements, then also u( (al),a 2 , . . . , a,) = (O), where (al)is the ideal generated in A by the element a,. PROOF. Let u = xlx2 and a be an arbitrary element of A . By (1 1) arid (12) a(ala2)= 0 =. (aal)a2= 0 and (a2al)a= 0 =. a2(ala) = 0 * (ala)a2= 0. Repeating this argument we obtain (al) u2 = (0). We carry out an induction on the length of the word u. Let d(u) = n > 2. We consider one of the cases that arises here: u = (ulu2)u3.If x, appears in u 3 , then by the induction assumption there is nothing to prove. Let x1 appear in u1 (the argument for u2 is analogous). The induction assumption allows us to assume that d(u2) = d(u3) = 1. Let u2 = x , - ~ , u3 = x,. Then 0 = u(a1, . . . ,a,) = ( U l ( U 1 ,
. . . ,an-2)an-1)u, = u,(a,, . . . ,U n - 2 ) ( U , -
la,).
By the induction assumption, (0)= ul((al),%, . . * = u((al),a2,*
-
9
9 4
..
~ , - 2 ) ( ~ , - l ~ ,= ) (Ul((al),a2,.
9
a,-2)afl-l)%
3
which is what was to be proved. LEMMA 4. If u(al, . . . , a,) = 0 for some multilinear nonassociative word u(xl, . . . , x,) and elements a l , . . . ,a, of an associative modulo zero
16.2.
NIL-ALGEBRAS
347
algebra A without nilpotent elements, then for any nonassociative word u(x,,. . . ,x,) of the same type as u it is also true u(a,, . . . , a,) = 0. PROOF. u = u ( a , , . . . , a,) is in the ideal (ai) for any i = 1, 2 , . . . , n. Therefore 0 = u(U,U, . . . , ii) = 2'' by Lemma 3, whence also follows the assertion.
THEOREM 3 (Ryabukhin). An algebra A is a subdirect sum of algebras without divisors of zero if and only if A is an associative modulo zero algebra without nilpotent elements.
PROOF. Let us assume that A is associative modulo zero and does not have nilpotent elements. Then the multiplicative subgroupoid generated by a nonzero element a E A does not contain zero, and by Zorn's lemma it is contained in some maximal subgroupoid which does not contain zero. We denote this groupoid by G,. We shall prove that I , = A\G, is an ideal in A . We show that the set I , is closed with respect to subtraction. Let x E I,. Then the subgroupoid ( G , , x) contains 0,and consequently there exists a nonassociative word u(x,, . . . , x,) and elements h , , h l , . . . ,h, E (G,,x), not all belonging to G,, such that u ( h l , . . . , h,) = 0. Hence by Lemma 4 it follows that x'g = 0 for some i 2 1 and g E G,. By that same lemma x'g = 0 implies (xg)' = 0, whence we obtain xg = 0. If y is another element from I , , then there can be found for it a g' E G, such that yg' = 0. By Lemma 2 we have gxg' = gyg' = 0,whence g(x - y)g' = 0. But this means that x - y 4 G,, that is, x - y E l a . Moreover, from Lemma 3 it follows that x E I, implies (x)c I,, so that I , is an ideal in A . As is easy to see, the algebra A / I , does not contain divisors of zero and a 4 I,. It is obvious that the algebra A is a subdirect sum of the algebras A / I , , which proves the theorem in one direction. The proof in the other direction is clear. COROLLARY (L'uou). Every alternative algebra without nilpotent elements is a subdirect sum of algebras without divisors of zero. Andrunakievich and Ryabukhin proved this theorem earlier for associative algebras. 2. NIL-ALGEBRAS
We consider the free right alternative algebra RA[X] from the set of free generators X = {x,,x2, . . , ). We shall call an element of the algebra RA[X]
16.
348
RIGHT ALTERNATIVE ALGEBRAS
a right alternative j-polynomial if it is expressible from elements of the set X by means of addition, multiplication by elements from 0,squaring, and the "quadratic multiplication" X U , = (yx)y. We denote by j R A [ x ] the set of all right alternative j-polynomials. If is the canonical homomorphism of the algebra RA[X] onto the free associative algebra Ass[X], then it is obvious .(jKA[xl) =j[Xl. LEMMA 5. Let f
=f(x,,.
Rf(x,*
. . , x,)
. . . ,x,)
=
EjRA[X]. Then
f "(Rx,,. . .
3
RXJ.
The proof repeats, verbatim, the proof of Lemma 5.12, where only the right alternative and right Moufang identities are used. Let A be a right alternative algebra and M = { m i } be a subset in A . We denote by ~ R A [ Mthe ] set of elements of the form f ( m , , . . . , mk) where f E jRA[x]. We shall call elements of the set j,,[M] j-polynomials from elements of the set M . LEMMA 6. In the right alternative @-algebra A let all j-polynomials from elements of some finite set M = {al,a,, . . . ,ak} be nilpotent, and in addition let their indices of nilpotency be bounded overall. Then the subalgebra M* of the algebra of right multiplications which is generated by the operators R,,, R,,, . . . , R,, is nilpotent. PROOF. Let J E j R A [ x ]
and f m ( a l , .. . , ak)= 0. Then by Lemma 5
(f")"(Ra,, .
3
Rak)
= R f m ( a , , . . . , a k ) = 0.
Thus all j-polynomials from R,,, . . . , R,, are nilpotent with an overall bound on the indices of nilpotency, whence by the Corollary to Theorem 5.3 follows the nilpotency of the subalgebra M*. This proves the lemma. We shall say that a subset M of an algebra A is right nilpotent if for some number N all r,-words from elements of the set M with length N (and consequently, also greater length) equal zero. An algebra is called locally r,-nilpotent if any finite subset of it is right nilpotent. THEOREM 4 (Shirshou). Every right alternative nil-algebra bounded index is locally r,-nilpotent.
of
The proof consists of applying Lemma 6 . This theorem has a series of important corollaries. COROLLARY 1. Every finite-dimensional right alternative nil-algebra
A over an arbitrary field @ is right nilpotent.
16.2.
349
NIL-ALGEBRAS
PROOF. Let a E A and n be the index of nilpotency of the element a. The chain of subspaces A
3
AR,
3
AR,I
3
. . . 3 AR,,
(0)
has length no greater than the dimension of the algebra A , and consequently, the indices of nilpotency of all the elements are bounded overall by the number 1 + dim, A . By Theorem 4 the algebra A is locally r,-nilpotent. In particular, its basis is right nilpotent. But this means that the algebra itself is also right nilpotent. COROLLARY 2. Every simple finite-dimensional right alternative algebra over a field is alternative. PROOF. By Corollary 1 a simple finite-dimensional right alternative algebra cannot be a nil-algebra. Consequently, it does not contain nil-ideals, and by the theorem of Albert which we mentioned in Section 1 it is alternative. It is natural to ask whether or not every finite-dimensional right alternative nil-algebra is also nilpotent? It turns out that they are not. The following example of a five-dimensional right nilpotent but not nilpotent algebra belongs to Dorofeev [SS]. Its basis is {a, b, c , d , e ) ,and the multiplication is given by the table (zero products of basis vectors are omitted)
ab = -ba
= ae =
-ea = db = -bd = -c,
ac = d,
bc = e.
The assertion of Corollary 1 can be strengthened. As shown by Shestakov, every right alternative @-algebra which is finitely generated as a @module by nilpotent generating elements is right nilpotent. The property of local r,-nilpotency was studied by Mikheev. He proved that this is a radical property in the sense of Amitsur-Kurosh, and that it is different from the usual local right nilpotency, which is not a radical property [151]. Exercises 1. (Shirshou) We define an operation ( ) on nonassociative words of the free algebra @[XI which cancels the arrangement of parentheses possessed by a nonassociative word and distributes them anew in the standard fashion from the right. For example: ( x 3 ((x,x4)x2)) = ( (x3x1)x4)x2. We also linearly extend the operation ( ) to polynomials. Let T R A be the T-ideal of the variety of right alternative algebras. Prove that ( T R A ) = (O), and by the same token show that the operation ( ) is reasonably defined for
16.
350
RIGHT ALTERNATIVE ALGEBRAS
the free right alternative algebra RA[X] = @[x]/TRA. Prove that for any j-polynomial f E jRA[X]we have f = ( f ) . 2. (Mikheeu) Prove that for any natural numbers n and t there exists a natural number M = M(n, t ) such that for any M elements x i l ,xi!, . . . , xiM ~i {xl, . . . , x,} G X in the right multiplication algebra of the free right alternative algebra RA[X] the followingequality holds:
wherejk(x)is a j-polynomial of degree 2 t. 3. (Mikheeu) If an ideal I and the quotient algebra A / I of a right alternative algebra A are locally r,-nilpotent, then A is also locally rl-nilpotent. Prove this assertion and deduce from this the existence of a locally r l nilpotent radical in the class of right alternative algebras. Hint. Use Exercises 1 and 2. 4. (Skosyrskiy) Prove that a right alternative algebra A is locally r l nilpotent if and only if the algebra A ( + )is locally nilpotent. 5. Prove that in every right alternative algebra are valid the relations
+
w(x 0y)' = (w 0x)y' (w 0y)x' - {xwy}, wx'y' + w(xy)' = 2(w 0x)y' + 2(w 0y)x' + 2w 0( x y ) - 4(w 0y) 0x, where u' is the Smiley notation for the operator of right multiplication R,. 6. Let A be a right alternative algebra over a ring (D containing f, and let I, be the right ideal of A generated by the set (A"))", where the power is understood with respect to the operation of multiplication in the algebra A ( + ) .Prove that for any elements al, a 2 , a3, a4 E A l,a\a;a;ak
c I,,,+ I .
Hint. Use the relations of Exercise 5. 7 . (Skosyrskiy) A right alternative algebra A over a ring @ containing f is right nilpotent if and only if the associated Jordan algebra A ( + )is nilpotent. Hint. Use the previous exercise. 8. Prove that if I is an ideal of a right alternative algebra A, then A1 is also an ideal in A. 9. (Slin'ko) Prove that a left and right nilpotent right alternative algebra is nilpotent. 10. Prove that for each natural number k there exists a natural number h(n, k) such that in the right multiplication algebra of the free right alternative algebra RA[x,, . . . ,xn] any word of degree h(n, k) is representable in the form of a linear combination of words each of which contains the operator R,,,,, where u ( x ) is a monomial of degree 2 k . Hint. See the proof of Lemma 4.2.
LITERATURE
35’
11. (Slin’ko) Every left nilpotent right alternative algebra with a finite number of generators is nilpotent. Hint. Use Exercises 8-10.
LITERATURE Albert [3, 71, Andrunakievich and Ryabukhin [19], Dorofeev [58], Kleinfeld [91,98,99], McCrimmon [138], Mikheev [147-1521, Pchelintsev [174]. Ryabukhin [187], Skornyakov [191, 1921, Skosyrskiy [194], Slin’ko [207], Smiley [218], Thedy [232-2341, Hentzel [245], Humm [255], Shestakov [267], Shirshov [276,279].
16
Right Alternative Algebras
Along with alternative algebras, there are often encountered in mathematics algebras with weaker conditions of associativity.The most important of these conditions is power-associativity,that is, every subalgebra generated by a single element is associative. The study of this broad class of algebras (in particular, it contains all anticommutative algebras) meets with insurmountable obstacles. However, some of its subclasses, which also contain the alternative algebras, have been adequately studied. Right alternative algebras, which arose from the study of a certain class of projective planes, have been studied the most intensively. In this chapter we shall briefly set forth the basic facts which are known up to the present time about right alternative algebras.
1. ALGEBRAS WITHOUT NILPOTENT ELEMENTS
An algebra is called right alternative if it satisfies the identities
34"
16.1.
343
ALGEBRAS WITHOUT NILPOTENT ELEMENTS
We recall that (1) is called the right alternative identity, and (2) is the right Moufang identity. By Proposition 1.4 the linearizations of these identities
+ ( X 4 Y = X ( Y " 4, ( ( X Y ) Z ) t + ( ( X W Y = x ( ( y z ) t + (tzly),
(1')
(XY)Z
(2')
are also valid in every right alternative algebra. If the algebra is without elements of additive order 2, then the right Moufang identity is a consequence of the right alternative identity. In fact, we have 2(yz)y = (z y ) y - z y2, whence by (1') and (1) 0
0
0
Y l - x(z Y 2 ) = [.(Z Y)lY + ( X Y ) ( Z Y ) - ( X Z ) Y 2 - ( X Y 2 ) Z = C(XZ)YIY + [(XY)ZIY + [ ( X Y ) Z l Y + C(XY)YlZ
2XC(YZ)Yl = .[(Z
O
Y)
O
O
O
- ( X Z ) Y 2 - (XY%
= 2[(XY)ZlY.
We note that the right Moufang identity is equivalent to the identity ( x , Y Z , Y ) = (x, z, Y)Y.
(3)
Actually, we have
0 = [ ( X Y ) Z I Y - X [ ( Y Z ) Y ] = ( x , Y, Z)Y
+ ( x , YZ7 Y )
= ( x , Y Z , Y ) - (x, z, Y)Y.
THEOREM 1. Every right alternative algebra is power-associative. PROOF. It suffices to prove that for arbitrary nonassociative words
ui = ui(x), where i = 1,2, 3, and an arbitrary element a of a right alternative algebra A, it is true (fi,,ii27fi3)= 0 where tii = ui(a).We shall carry out the
proof by induction on the total length d(u,) in any algebra there is the equality ( X Y , 2, t )
- (x, Y Z , r ) + (x, Y,
+ d(u2)+ d(u3). By the fact that
= X ( Y , z, t )
+ (x, Y ,z ) ~ ,
we can assume that u3(x)= x. Furthermore, by the induction assumption one can assume that u2 = x . u ; . It is now clear that in view of (3), (El,E2,fi3)
= @,,a
.ii;,u)
= 0.
This proves the theorem. With a view toward a compact presentation of calculations, we shall make use of notation which was introduced into use by Smiley. We denote by c' the operator of right multiplication R,. We then set cd = (cd)' - c'd' and
344
16.
RIGHT ALTERNATIVE ALGEBR /is
cd = (cd)’ - d’c’. Identities (I), (l’),(2), and (2’) imply the relations
+
(cd + dc)’ = c’d’ d’c‘, c‘ = 0, cd d‘ = 0, c’d‘c‘ = [ ( c d ) ~ ] ’ , c‘d‘s’ s’d’c’ = [ ( c d ) ~ ( s d ) ~ ] ’ . c’c‘ = (c2)1,
+
+
+
THEOREM 2 (Mikheeu). In every right alternative algebra the (Mikheev) identity (x, x, YI4 = 0
is valid. PROOF. Let a, b, c be arbitrary elements of a right alternative algebra A. We set q = [a, b] and p = (a,a, b). We shall prove the relations
abab =
aba‘ab =
- (4,a, b)’, - (qa, a, b)’,
Relation (4) is obvious. Moreover,
+
(ab)’(ab)’- (ab)‘b’a’- a’b’(ab)’ a’b’ba‘ = [(ab)(ab)]’- [((ab)b)a (ab)(ab)]‘ [(ab2)a]’= 0.
abab =
+
+
The second part of ( 5 ) is linearization of the first. We shall now prove (’7):
+ +
aba‘ab = (ab)’d(ab)‘- (ab)’a’a’b’- b’a’a’(ab)’ ba’a‘a‘b‘ = ([(ab)a](ab) - [(ab)a2]b- (ba2)(ab) (ba3)b}‘ = - (qa,a, b)’.
Relation (6) is proved analogously. Now by (4),(5), (6), and (7) p(4, a, b) =
a, b)’ = aababab= 0,
- aab(q,
P(qa, a, b) =
a, b)’ = aababdab= 0.
In addition, we have the relations p‘ = abo- dab = abo - aha'.
We shall only prove the first: p‘ = -(a, b, a)’ = [a(ba)- (&)a]’ = [ ~ ( b a ) ]-‘ a‘b’a‘ = [ ~ ( b a ) ] ’ da’b‘ - a’(ba)’ - a’(ab)’ = abU- dab.
+
(8) (9)
16.1.
ALGEBRAS WITHOUT NILPOTENT ELEMENTS
345
COROLLARY 1 (Kleinfeld). Every right alternative algebra without nilpotent elements is alternative.
A much stronger assertion, due to Albert [3, 71, is valid for finitedimensional algebras. He proved that every finite-dimensional right alternative algebra over a field of characteristic 2 2 , which does not contain nil-ideals, is alternative. The proof is technically complicated, and we shall not produce it here. COROLLARY 2 (Skornyakou). Every right alternative division algebra is alternative. Skornyakov proved by means of this result that the realization of little Desargues’ theorem on two lines of a projective plane implies its projective realization in that plane. The question on the structure of simple right alternative algebras is at present not clear to its conclusion. Thedy proved that under certain additional restrictions, simple right alternative algebras are alternative [232, 2331. For example, this is so if the simple algebra A contains an idempotent e # 0, 1 for which (e,e, A ) = (0).There was the conjecture that simple right alternative algebras which are not alternative do not exist. However, Mikheev [152] recently disproved this conjecture by constructing an appropriate example. It is interesting that the algebra in Mikheev’s example satisfies the identity x3 = 0, that is, is a nil-algebra of bounded index. The hope, nevertheless, remains that right alternative algebras which do not contain nil-ideals are alternative. Some basis for such a conjecture is the above-mentioned result of Albert. We now study in somewhat more detail alternative(and consequently, also right alternative) algebras without nilpotent elements. LEMMA 1. In every alternative algebra without nilpotent elements
(ab)c = 0 0 a(bc) = 0.
(1 1)
16.
346
RIGHT ALTERNATIVE ALGEBRAS
PROOF. If (ab)c = 0, then using the middle Moufang identity we have
[ ~ ( b c )=] a(bc ~ - (U . bc . a) * bc) = a(bc . (ab * CU) * bc) = a((bc * ab)(ca . bc)) = a((bc . ab)(c . ab . c ) ) = 0, whence a(bc) = 0. The reverse implication is proved analogously. Algebras satisfying condition (1 1) are called associatiue modulo zero. They were introduced into consideration by Ryabukhin. It is easy to see that if in an associative modulo zero algebra a product of some n elements equals zero for one arrangement of parentheses, then it also equals zero for any other arrangement. LEMMA 2. In every associative modulo zero algebra without nilpotent elements ab = 0
ba = 0.
(12)
PROOF. ab = 0 * b(ab)= 0 * (ba)b= 0 => [(ba)b]a= 0 ba = 0.
(ba)2= 0
LEMMA 3. Let u = u(xl, . . . ,x,) be a multilinear nonassociative word. If u(al,u2, . . . , a,) = 0 for elements a,, a 2 , . . . , a, of an associative modulo zero algebra A without nilpotent elements, then also u( (al),a 2 , . . . , a,) = (O), where (al)is the ideal generated in A by the element a,. PROOF. Let u = xlx2 and a be an arbitrary element of A . By (1 1) arid (12) a(ala2)= 0 =. (aal)a2= 0 and (a2al)a= 0 =. a2(ala) = 0 * (ala)a2= 0. Repeating this argument we obtain (al) u2 = (0). We carry out an induction on the length of the word u. Let d(u) = n > 2. We consider one of the cases that arises here: u = (ulu2)u3.If x, appears in u 3 , then by the induction assumption there is nothing to prove. Let x1 appear in u1 (the argument for u2 is analogous). The induction assumption allows us to assume that d(u2) = d(u3) = 1. Let u2 = x , - ~ , u3 = x,. Then 0 = u(a1, . . . ,a,) = ( U l ( U 1 ,
. . . ,an-2)an-1)u, = u,(a,, . . . ,U n - 2 ) ( U , -
la,).
By the induction assumption, (0)= ul((al),%, . . * = u((al),a2,*
-
9
9 4
..
~ , - 2 ) ( ~ , - l ~ ,= ) (Ul((al),a2,.
9
a,-2)afl-l)%
3
which is what was to be proved. LEMMA 4. If u(al, . . . , a,) = 0 for some multilinear nonassociative word u(xl, . . . , x,) and elements a l , . . . ,a, of an associative modulo zero
16.2.
NIL-ALGEBRAS
347
algebra A without nilpotent elements, then for any nonassociative word u(x,,. . . ,x,) of the same type as u it is also true u(a,, . . . , a,) = 0. PROOF. u = u ( a , , . . . , a,) is in the ideal (ai) for any i = 1, 2 , . . . , n. Therefore 0 = u(U,U, . . . , ii) = 2'' by Lemma 3, whence also follows the assertion.
THEOREM 3 (Ryabukhin). An algebra A is a subdirect sum of algebras without divisors of zero if and only if A is an associative modulo zero algebra without nilpotent elements.
PROOF. Let us assume that A is associative modulo zero and does not have nilpotent elements. Then the multiplicative subgroupoid generated by a nonzero element a E A does not contain zero, and by Zorn's lemma it is contained in some maximal subgroupoid which does not contain zero. We denote this groupoid by G,. We shall prove that I , = A\G, is an ideal in A . We show that the set I , is closed with respect to subtraction. Let x E I,. Then the subgroupoid ( G , , x) contains 0,and consequently there exists a nonassociative word u(x,, . . . , x,) and elements h , , h l , . . . ,h, E (G,,x), not all belonging to G,, such that u ( h l , . . . , h,) = 0. Hence by Lemma 4 it follows that x'g = 0 for some i 2 1 and g E G,. By that same lemma x'g = 0 implies (xg)' = 0, whence we obtain xg = 0. If y is another element from I , , then there can be found for it a g' E G, such that yg' = 0. By Lemma 2 we have gxg' = gyg' = 0,whence g(x - y)g' = 0. But this means that x - y 4 G,, that is, x - y E l a . Moreover, from Lemma 3 it follows that x E I, implies (x)c I,, so that I , is an ideal in A . As is easy to see, the algebra A / I , does not contain divisors of zero and a 4 I,. It is obvious that the algebra A is a subdirect sum of the algebras A / I , , which proves the theorem in one direction. The proof in the other direction is clear. COROLLARY (L'uou). Every alternative algebra without nilpotent elements is a subdirect sum of algebras without divisors of zero. Andrunakievich and Ryabukhin proved this theorem earlier for associative algebras. 2. NIL-ALGEBRAS
We consider the free right alternative algebra RA[X] from the set of free generators X = {x,,x2, . . , ). We shall call an element of the algebra RA[X]
16.
348
RIGHT ALTERNATIVE ALGEBRAS
a right alternative j-polynomial if it is expressible from elements of the set X by means of addition, multiplication by elements from 0,squaring, and the "quadratic multiplication" X U , = (yx)y. We denote by j R A [ x ] the set of all right alternative j-polynomials. If is the canonical homomorphism of the algebra RA[X] onto the free associative algebra Ass[X], then it is obvious .(jKA[xl) =j[Xl. LEMMA 5. Let f
=f(x,,.
Rf(x,*
. . , x,)
. . . ,x,)
=
EjRA[X]. Then
f "(Rx,,. . .
3
RXJ.
The proof repeats, verbatim, the proof of Lemma 5.12, where only the right alternative and right Moufang identities are used. Let A be a right alternative algebra and M = { m i } be a subset in A . We denote by ~ R A [ Mthe ] set of elements of the form f ( m , , . . . , mk) where f E jRA[x]. We shall call elements of the set j,,[M] j-polynomials from elements of the set M . LEMMA 6. In the right alternative @-algebra A let all j-polynomials from elements of some finite set M = {al,a,, . . . ,ak} be nilpotent, and in addition let their indices of nilpotency be bounded overall. Then the subalgebra M* of the algebra of right multiplications which is generated by the operators R,,, R,,, . . . , R,, is nilpotent. PROOF. Let J E j R A [ x ]
and f m ( a l , .. . , ak)= 0. Then by Lemma 5
(f")"(Ra,, .
3
Rak)
= R f m ( a , , . . . , a k ) = 0.
Thus all j-polynomials from R,,, . . . , R,, are nilpotent with an overall bound on the indices of nilpotency, whence by the Corollary to Theorem 5.3 follows the nilpotency of the subalgebra M*. This proves the lemma. We shall say that a subset M of an algebra A is right nilpotent if for some number N all r,-words from elements of the set M with length N (and consequently, also greater length) equal zero. An algebra is called locally r,-nilpotent if any finite subset of it is right nilpotent. THEOREM 4 (Shirshou). Every right alternative nil-algebra bounded index is locally r,-nilpotent.
of
The proof consists of applying Lemma 6 . This theorem has a series of important corollaries. COROLLARY 1. Every finite-dimensional right alternative nil-algebra
A over an arbitrary field @ is right nilpotent.
16.2.
349
NIL-ALGEBRAS
PROOF. Let a E A and n be the index of nilpotency of the element a. The chain of subspaces A
3
AR,
3
AR,I
3
. . . 3 AR,,
(0)
has length no greater than the dimension of the algebra A , and consequently, the indices of nilpotency of all the elements are bounded overall by the number 1 + dim, A . By Theorem 4 the algebra A is locally r,-nilpotent. In particular, its basis is right nilpotent. But this means that the algebra itself is also right nilpotent. COROLLARY 2. Every simple finite-dimensional right alternative algebra over a field is alternative. PROOF. By Corollary 1 a simple finite-dimensional right alternative algebra cannot be a nil-algebra. Consequently, it does not contain nil-ideals, and by the theorem of Albert which we mentioned in Section 1 it is alternative. It is natural to ask whether or not every finite-dimensional right alternative nil-algebra is also nilpotent? It turns out that they are not. The following example of a five-dimensional right nilpotent but not nilpotent algebra belongs to Dorofeev [SS]. Its basis is {a, b, c , d , e ) ,and the multiplication is given by the table (zero products of basis vectors are omitted)
ab = -ba
= ae =
-ea = db = -bd = -c,
ac = d,
bc = e.
The assertion of Corollary 1 can be strengthened. As shown by Shestakov, every right alternative @-algebra which is finitely generated as a @module by nilpotent generating elements is right nilpotent. The property of local r,-nilpotency was studied by Mikheev. He proved that this is a radical property in the sense of Amitsur-Kurosh, and that it is different from the usual local right nilpotency, which is not a radical property [151]. Exercises 1. (Shirshou) We define an operation ( ) on nonassociative words of the free algebra @[XI which cancels the arrangement of parentheses possessed by a nonassociative word and distributes them anew in the standard fashion from the right. For example: ( x 3 ((x,x4)x2)) = ( (x3x1)x4)x2. We also linearly extend the operation ( ) to polynomials. Let T R A be the T-ideal of the variety of right alternative algebras. Prove that ( T R A ) = (O), and by the same token show that the operation ( ) is reasonably defined for
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RIGHT ALTERNATIVE ALGEBRAS
the free right alternative algebra RA[X] = @[x]/TRA. Prove that for any j-polynomial f E jRA[X]we have f = ( f ) . 2. (Mikheeu) Prove that for any natural numbers n and t there exists a natural number M = M(n, t ) such that for any M elements x i l ,xi!, . . . , xiM ~i {xl, . . . , x,} G X in the right multiplication algebra of the free right alternative algebra RA[X] the followingequality holds:
wherejk(x)is a j-polynomial of degree 2 t. 3. (Mikheeu) If an ideal I and the quotient algebra A / I of a right alternative algebra A are locally r,-nilpotent, then A is also locally rl-nilpotent. Prove this assertion and deduce from this the existence of a locally r l nilpotent radical in the class of right alternative algebras. Hint. Use Exercises 1 and 2. 4. (Skosyrskiy) Prove that a right alternative algebra A is locally r l nilpotent if and only if the algebra A ( + )is locally nilpotent. 5. Prove that in every right alternative algebra are valid the relations
+
w(x 0y)' = (w 0x)y' (w 0y)x' - {xwy}, wx'y' + w(xy)' = 2(w 0x)y' + 2(w 0y)x' + 2w 0( x y ) - 4(w 0y) 0x, where u' is the Smiley notation for the operator of right multiplication R,. 6. Let A be a right alternative algebra over a ring (D containing f, and let I, be the right ideal of A generated by the set (A"))", where the power is understood with respect to the operation of multiplication in the algebra A ( + ) .Prove that for any elements al, a 2 , a3, a4 E A l,a\a;a;ak
c I,,,+ I .
Hint. Use the relations of Exercise 5. 7 . (Skosyrskiy) A right alternative algebra A over a ring @ containing f is right nilpotent if and only if the associated Jordan algebra A ( + )is nilpotent. Hint. Use the previous exercise. 8. Prove that if I is an ideal of a right alternative algebra A, then A1 is also an ideal in A. 9. (Slin'ko) Prove that a left and right nilpotent right alternative algebra is nilpotent. 10. Prove that for each natural number k there exists a natural number h(n, k) such that in the right multiplication algebra of the free right alternative algebra RA[x,, . . . ,xn] any word of degree h(n, k) is representable in the form of a linear combination of words each of which contains the operator R,,,,, where u ( x ) is a monomial of degree 2 k . Hint. See the proof of Lemma 4.2.
LITERATURE
35’
11. (Slin’ko) Every left nilpotent right alternative algebra with a finite number of generators is nilpotent. Hint. Use Exercises 8-10.
LITERATURE Albert [3, 71, Andrunakievich and Ryabukhin [19], Dorofeev [58], Kleinfeld [91,98,99], McCrimmon [138], Mikheev [147-1521, Pchelintsev [174]. Ryabukhin [187], Skornyakov [191, 1921, Skosyrskiy [194], Slin’ko [207], Smiley [218], Thedy [232-2341, Hentzel [245], Humm [255], Shestakov [267], Shirshov [276,279].