Non-associative rings and algebras

Sets with two binary operations $+$ and $\cdot$, satisfying all the axioms of associative rings and algebras except possibly the associativity of multiplication. The first examples of non-associative rings and algebras that are not associative appeared in the mid-19th century (Cayley numbers and, in general, hypercomplex numbers, cf. Hypercomplex number). Given an associative ring (algebra), if one replaces the ordinary multiplication by the operation $[a,b] = ab-ba$, the result is a non-associative ring (algebra) that is a Lie ring (algebra). Yet another important class of non-associative rings (algebras) is that of Jordan rings (algebras); these are obtained by defining the operation $a \cdot b = (ab+ba)/2$ in an associative algebra over a field of characteristic $\neq 2$ (or over a commutative ring of operators with a 1 and a $1/2$). The theory of non-associative rings and algebras has evolved into an independent branch of algebra, exhibiting many points of contact with other fields of mathematics and also with physics, mechanics, biology, and other sciences. The central part of the theory is the theory of what are known as nearly-associative rings and algebras: Lie, alternative, Jordan, Mal'tsev rings and algebras, and some of their generalizations (see Lie algebra; Alternative rings and algebras; Jordan algebra; Mal'tsev algebra).

One of the most important problems that must be solved when studying any class of non-associative algebras is the description of simple algebras, both finite dimensional and infinite dimensional. In this context, the word description is to be understood modulo some "classical" class contained in the class being described (e.g. the description of simple algebras in the class of alternative rings is given modulo associative rings; for Mal'tsev algebras — modulo Lie algebras; for Jordan algebras — modulo special Jordan algebras; etc.). From this point of view, the various classes of non-associative algebras can be divided into those in which there are "many" simple algebras and those in which there are "few" . Typical classes in which there are many simple algebras are the associative algebras, the Lie algebras and the special Jordan algebras. Namely, in these classes the following imbedding theorem is valid: Any associative (Lie, special Jordan) algebra over a field can be imbedded in a simple algebra of the same type. In some classes of algebras there are many simple algebras that are far from associative — in the class of all algebras and in the class of all commutative (anti-commutative) algebras. For these classes, too, there holds an imbedding theorem analogous to that cited above.

The problem of describing the finite-dimensional simple associative (Lie, alternative or Jordan) algebras is the object of the classical part of the theory of these algebras. Subsequently, the main results about the structure of simple finite-dimensional associative (alternative, Jordan) algebras were carried over to Artinian rings of the same type — rings with the minimum condition for one-sided ideals; in Jordan rings, one-sided ideals are replaced by quadratic ideals (see Jordan algebra).

Classes of algebras with "few" simple algebras are interesting. Typical examples are the classes of alternative, Mal'tsev or Jordan algebras. In the class of alternative algebras, modulo associative algebras the only simple algebras are the (eight-dimensional) Cayley–Dickson algebras over an associative-commutative centre. In the class of Mal'tsev algebras, modulo Lie algebras the only simple algebras are the (seven-dimensional) algebras (relative to the commutator operation $[a,b]$) associated with the Cayley–Dickson algebras. In the class of Jordan algebras, modulo the special Jordan algebras the simple algebras are the (twelve-dimensional) Albert algebras over their associative centres (algebras of the series $E$) (see Jordan algebra). In larger classes, such as those of right-alternative or binary Lie algebras, the description of simple algebras is as yet incomplete (1989). It is known that there exists no finite-dimensional simple binary Lie algebra over a field of characteristic 0 other than a Mal'tsev algebra, but it is not known whether this result is valid in the infinite-dimensional case. For right-alternative algebras it is known that, although all finite-dimensional simple algebras of this class are alternative, there exist infinite-dimensional simple right-alternative algebras that are not alternative. All simple algebras are associative for the so-called $(\gamma,\delta)$-algebras (provided $(\gamma,\delta) \neq (1,1)$); these algebras arise in a natural manner from the stipulation that the square of an ideal be an ideal. A description is known for all Jordan algebras with two generators: Any Jordan algebra with two generators is a special Jordan algebra (Shirshov's theorem). All Jordan division algebras have been described (modulo associative division algebras).

In the classes of alternative, Mal'tsev or Jordan algebras there is a description of all primary rings (i.e. algebras the groupoid of two-sided ideals of which does not contain a zero divisor), as follows. A primary alternative ring (with $1/3$ in the commutative ring of operators) is either associative or a Cayley–Dickson ring. A primary non-degenerate Jordan algebras is either special or is an Albert ring (a Jordan ring is called an Albert ring if its associative centre $Z$ consists of regular elements and if the algebra $Z^{-1}A$ is a twenty-seven-dimensional Albert algebra over its centre $Z^{-1}Z$).

In a certain sense, the opposite of a simple algebra or a primary algebra is a nil algebra. For power-associative algebras (cf. Algebra with associative powers) that are not anti-commutative (such as associative, alternative, Jordan, etc., algebras), nil algebras are defined as algebras in which some power of each element equals zero; in the case of anti-commutative algebras (i.e. algebras with the identity $x^2=0$, such as Lie, Mal'tsev and binary Lie algebras), nil algebras are the same as Engel algebras, i.e. algebras satisfying a condition $$ \forall x,y \exists \overbrace{((x y) \cdots y)}^{n} = 0 \ . $$

In alternative (including associative) algebras, any nil algebra of bounded index (i.e. with an identity $x^n = 0$) is locally nilpotent, and if it has no $m$-torsion (i.e. $mx = 0 \Rightarrow x=0$) for $m \le n$, it is solvable (in the associative case — nilpotent). Shirshov's problem concerning the local nilpotency of Jordan nil algebras of bounded index has been solved affirmatively. It is not known (1989) whether there exists a simple associative nil ring.

In the case of Lie algebras, the problem of the local nilpotency of Engel Lie algebras is solved by Kostrikin's theorem: Any Lie algebra with an identity $$ \overbrace{[\ldots[x,y], \ldots ,y]}^{n} = 0 \ . $$ over a field of characteristic $p>n$ is locally nilpotent. This theorem implies a positive solution to the restricted Burnside problem for groups of exponent $p$. Recently, E.I. Zel'manov (1989) has proved the local nilpotency of Engel Lie algebras over a field of arbitrary characteristic. From this he has inferred a positive solution of the restricted Burnside problem for groups of arbitrary exponent $n$ (using the classification of the finite simple groups). In general, all problems connected with the local nilpotency of nil algebras are known as Burnside-type problems. Among these is also Kurosh problem concerning the local finiteness of algebraic algebras (cf. Algebraic algebra). An alternative (in particular, associative) algebraic algebra $A$ of bounded degree (i.e. the degrees of the polynomials satisfied by elements of $A$ are uniformly bounded) is locally finite. In the general case, however, Burnside-type problems (such as the local nilpotency of associative nil rings, etc.) have negative solutions.

Another topic of study includes free algebras and free products of algebras in various varieties. In the variety of all non-associative algebras, any subalgebra of a free algebra is free, and any subalgebra of a free product of algebras is the free product of its intersections with the factors and some free algebra (Kurosh theorem). Theorems of this type are also valid in varieties of commutative (anti-commutative) algebras. These questions are most interesting for Lie algebras. Any subalgebra of a free Lie algebra is itself a free Lie algebra (the Shirshov–Witt theorem). However, the analogue of Kurosh theorem is no longer valid for subalgebras of a free product of Lie algebras; nevertheless, such subalgebras may be described in terms of the generators of an ideal modulo which the free product of the intersections and the free subalgebra must be factorized. Research has been done on free alternative algebras — their Zhevlakov radicals (quasi-regular radicals, cf. Quasi-regular radical), their centres (associative and commutative), the quotient algebras modulo the Zhevlakov radical, etc. In contrast to free associative algebras, free alternative algebras with $n \ge 4$ generators contain zero divisors and, moreover, trivial ideals (non-zero ideals with zero square). There are also known instances of trivial ideals in free Mal'tsev algebras with $n \ge 5$ generators; while concerning free Jordan algebras with $n \ge 3$ generators all that is known is that they contain zero divisors, nil elements and central elements.

The theory of free algebras is closely bound up with questions of identities in various classes of algebras. In this connection one also has the problem of the basis rank of a variety (the basis rank is the smallest natural number $n$ such that the variety in question is generated by a free algebra with $n$ generators; if no such $n$ exists, the basis rank is defined as infinity). The basis rank of the varieties of associative and Lie algebras is 2; that of alternative and Mal'tsev algebras is infinite.

The general theory of varieties and classes of non-associative algebras deals with classes of algebras on the borderline of the classical ones and with their various relationships. One characteristic result is the following. It turns out that the varieties of admissible, generalized admissible and generalized standard algebras defined at different times and by different authors actually belong to the eight-element sublattice of the lattice of all varieties of non-associative algebras, which is also made up of the varieties of Jordan, commutative, associative, associative-commutative, and alternative algebras. The variety generated by a finite associative (alternative, Lie, Mal'tsev, or Jordan) ring is finitely based, while there exists a finite non-associative ring (an algebra over a finite field) that generates an infinitely based variety. There exists a Lie algebra over an infinite field with this property. At the same time, it is still (1989) not known whether there exists a non-finitely based variety of Lie algebras over a field of characteristic zero. A.R. Kemer [18] has proved that every variety of associative algebras over a field of characteristic 0 is finitely based (a positive solution to Specht's problem).

The algorithmic problems in the theory of non-associative rings and algebras have been formulated under the influence of mathematical logic. It is known that the word problem in the variety of all non-associative algebras is solvable (Zhukov's theorem). An analogous result is valid for commutative (anti-commutative) algebras. It is known that the Lie algebras with one relation have a solvable word problem. At the same time, there exist finitely-presented Lie algebras with an unsolvable word problem. The word problem has also been investigated in the variety of solvable Lie algebras of a given solvability degree $n$; it is solvable for $n=2$, unsolvable for $n \ge 3$. It has been proved that any recursively-defined Lie algebra (associative algebra) over a prime field can be imbedded in a finitely-presented Lie algebra (associative algebra).

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