Alternative rings and algebras

An alternative ring is a ring in which every two elements generate an associative subring; an alternative algebra is a (linear) algebra that is an alternative ring. By a theorem of >E. Artin the class of all alternative rings is defined by the system of identities:

$$ (xy)y = x(yy) \ ( \textrm { right } \textrm{ alternativeness } ) ; $$

$$ (xx)y = x(xy) \ ( \textrm { left } \textrm{ alternativeness } ) .$$

Thus, the alternative rings form a variety. The terminology "alternative ring" is justified by the fact that in any such ring the associator (the defect of associativity)

$$ ( x , y , z ) = ( x y ) z - x ( y z ) $$

is a skew-symmetric (alternative) function of its arguments.

The first example of alternative rings were the Cayley numbers, which form an alternative skew-field, i.e. an alternative ring with a unit element, in which the equations $ ax = b $ and $ ya = b $ are uniquely solvable for all $ b $ and all $ a \neq 0 $. Alternative skew-fields play a substantial part in the theory of projective planes, since a projective plane is a Moufang plane (i.e. a translation plane with respect to some straight line) if and only if any coordinatization of its ternary ring is an alternative skew-field. If in a ring $ R $ with a unit element each non-zero element is invertible, and if the identity

$$ a ^ {-1} ( a b ) = b $$

(or the identity $ (ba) a ^ {-1} = b $) is valid for any $ a, b \in R $, then $ R $ is an alternative skew-field. Any alternative skew-field is either associative or has the structure of a Cayley–Dickson algebra over its centre.

Each simple alternative ring is also either associative or a Cayley–Dickson algebra over its centre (in this case the algebra need not necessarily be a skew-field). Both the associative and the primitive alternative rings are exhausted by the Cayley–Dickson algebras. All primary alternative rings $ R $( if $ 3R \neq 0 $) are either associative or are Cayley–Dickson rings.

Many properties of alternative rings differ substantially from those of an associative ring in a similar situation. Thus, if $ R $ is an alternative ring and $ A $ and $ B $ are right ideals in it, their product $ AB $ need not necessarily be a right ideal, even if $ A $ is a two-sided ideal in $ R $; however, the product of two two-sided ideals of an alternative ring is a two-sided ideal in it. The difference between associative rings and alternative rings is also strongly manifested by the fact that alternative rings contain various kinds of nilpotency, since the product of elements may be zero or non-zero, depending on the placement of the parentheses. It is customary to use the following kind of nilpotency in an alternative ring: solvability (a ring $ R $ is called a solvable ring of index $ m $ if there exists a number $ m $ such that $ R _ {m} = 0 $, where $ R _ {i+1} = R _ {i} R _ {i} $, $ R _ {1} = R $), right nilpotency (there exists a number $ n $ such that $ R ^ {(n)} = 0 $, where $ R ^ {(i+1)} = R ^ {(i)} R ^ {(1)} $, $ R ^ {(1)} = R $), and nilpotency (there exists a number $ k $ such that $ R ^ {k} = 0 $, i.e. the product of any $ k $ elements of $ R $ is zero, no matter how the parentheses are placed). There are solvable alternative rings of index 3, but they are not nilpotent. Right nilpotency in an alternative ring is equivalent to nilpotency (alternative rings that are right nilpotent of index $ n $, are also nilpotent of index $ \leq {(n + 1) } ^ {2} $). Locally, i.e. in finitely generated rings, all kind of nilpotency are equivalent. The theory that establishes sufficient criteria of local nilpotency of an alternative ring is completely parallel to the corresponding theory for associative rings. This is a consequence of the following fact: Let $ R $ be an alternative ring in which one can to select a system of generators $ S $ such that any two elements of $ S $ generate a nil ring, suppose also that all associative homomorphic images of $ R $ are locally nilpotent, then $ R $ is locally nilpotent. Therefore, if $ R $ is an alternative ring with an identity $ x ^ {n} = 0 $, then $ R $ is locally nilpotent; if $ R $ is an alternative algebra with an identity relation that is not a consequence of the associativity, and if each element is the sum of a finite number of nil elements, then $ R $ is locally nilpotent. As regards global, as distinct from local, nilpotency the situation in alternative rings differs from that in associative rings. Thus, an alternative ring with the identity $ x ^ {3} = 0 $ is not necessarily nilpotent (even if its additive group is torsion-free). However, an alternative ring with an identity $ x ^ {n} = 0 $ and without elements of order $ k $, $ 0 < k \leq n $, in the additive group is solvable of index $ n(n + 1) / 2 $.

If $ R $ is an algebraic alternative algebra with an identity relation that is not a consequence of the associativity (or if the degrees of algebraicity of the elements of $ R $ are uniformly bounded), then $ R $ is locally finite-dimensional.

In a alternative ring there is an analogue of the Jacobson radical: In each alternative ring there exists a maximal quasi-regular ideal $ J(R) $, which is equal to the intersection of all modular maximal right ideals. The quotient ring $ R/J(R) $ is $ J $- semi-simple, i.e. $ J(R/J(R)) = 0 $; if $ I $ is an ideal of $ R $, then $ J(I) = J(R) \cap I $, and any $ J $- semi-simple ring can be approximated by primitive alternative rings (i.e. by primitive associative rings and by Cayley–Dickson algebras). There also exist analogues of all other associative radicals (lower nil-radical, locally nilpotent radical, etc.), which display the same basic properties as in associative rings.

In an Artinian alternative ring $ R $( i.e. an alternative ring that satisfies the descending chain condition (minimum condition) for right ideals) the radical $ R(J) $ is nilpotent, and so is any nil-subgroupoid of the multiplicative groupoid of $ R $. A ring $ R $ is an Artinian alternative ring without nilpotent ideals if and only if $ R $ splits into the direct sum of finitely many complete matrix algebras (over certain associative skew-fields) and Cayley–Dickson algebras; this decomposition is unique for each $ R $, apart from a rearrangement of the terms. If $ R $ is an alternative ring, $ I $ one of its ideals, and $ R $ satisfies the minimum condition for the two-sided ideals contained in $ I $, then $ I $ is nilpotent if and only if there are no simple ideals in the ring $ R \phi $ for any arbitrary homomorphism $ \phi $ from $ R $ into $ I \phi $.

In an alternative ring $ R $ one distinguishes between the associative centre

$$ N ( R ) = \{ {n \in R } : {( n , a , b ) = 0 \textrm{ for all } \ a , b \in R } \} , $$

the commutative centre

$$ C ( R ) = \{ {c \in R } : {[ c , a ] = ca - ac = 0 \ \textrm{ for all } \ a \in R } \} , $$

and the centre

$$ Z ( R ) = N ( R ) \cap C ( R ) . $$

If in the additive group of $ R $ there are no elements of order three, then $ C(R) \subseteq N(R) $. However, commutative non-associative alternative algebras exist over a field of characteristic three. In a primary alternative ring $ R $ one always has $ C(R) \supseteq N(R) $. In any alternative ring $ R $ one always has $ [ N(R), R] \subseteq N(R) $. Suppose that in an alternative ring $ R $ there are no non-trivial ideals. Then 1) either $ 3R = 0 $ or $ N(R) \neq 0 $; 2) either $ 3R \subseteq N(R) $ or $ Z(R) \neq 0 $; 3) if $ A $ is a right ideal in $ R $, then $ N(A) = A \cap N(R) $ and $ Z(A) = A \cap Z(R) $. However, over any field $ F $ it is possible to construct an alternative ring $ K $ with no non-trivial ideals such that $ N(K) = C(K) = 0 $, but $ K ^ { 2 } $, which is an ideal of $ K $, is an associative-commutative ring, i.e.

$$ N ( K ^ { 2 } ) = C ( K ^ { 2 } ) = K ^ { 2 } \neq 0 . $$

Of the identities that are valid in an alternative ring, the following are the best known:

$$ [ ( x y ) z ] y = x [( y z ) y ] , $$

$$ [ ( x y ) x ] z = x [ y ( x z ) ] , $$

$$ ( x y ) ( z x ) = [ x ( y z ) ] x $$

(the Moufang identities);

$$ ( x y , z , t ) - y ( x , z , t ) - ( y , z , t ) x = $$

$$ = \ ( [ x , y ] , z , t ) + ( x , y , [ z , t ] ) , $$

$$ ( [ x , y ] ^ {4} , z , t ) = [ x , y ] ( [ x , y ] ^ {2} , z , t ) = $$

$$ = \ ( [ x , y ] ^ {2} , z , t ) [ x , y ] = 0 . $$

An alternative ring with three generators also satisfies the identity

$$ \tag{* } ( [ x , y ] [ z , t ] + [ z , t ] [ x , y ] , u , v ) = 0 . $$

In an alternative ring with more than three generators the identity (*) is usually not satisfied; moreover, in these alternative rings, as a rule, $ ( [x, y] ^ {2} , z, t) \neq 0 $. In any alternative ring without locally nilpotent ideals, the identity (*) is satisfied, so that such an alternative ring can be approximated by primary associative rings and by Cayley–Dickson rings.

Any free alternative ring $ R $ has a non-zero ideal $ U(R) $ contained in the associative centre $ N(R) $. A free alternative ring with three or more generators not only contains divisors of zero, but is not primary. Free alternative rings with four or more generators even contain trivial ideals and for this reason cannot be approximated by primary rings.

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An element $ z $ in a ring $ R $ is right quasi-regular if $ \{ {a - z a } : {a \in R } \} = R $. An element is quasi-regular if it is both left and right quasi-regular. An ideal in $ R $ is quasi-regular if all its elements are quasi-regular [a2]. The phrase "approximated by …" means "representable as a subdirect sum of …" . For example "a J-semi-simple alternative algebra R can be approximated by primitive alternative rings" means that $ R $ is isomorphic to a subdirect sum of primitive alternative rings.

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