Nil algebra

A power-associative (in particular, an associative) algebra (cf. Algebra with associative powers) in which every element is nilpotent (cf. Nilpotent element). Special cases of nil algebras are nilpotent and locally nilpotent algebras (cf. Nilpotent algebra; Locally nilpotent algebra). In the associative case the construction of a nil algebra that is not locally nilpotent is a difficult problem; essentially only one example of such an algebra is known (see [5]).

The class of nil algebras is closed under taking subalgebras and homomorphic images. The extension of a nil algebra by a nil algebra is again a nil algebra. Therefore, in every algebra the sum of all nil ideals is the largest nil ideal containing every nil ideal. It is called the upper nil radical of the algebra, and its quotient algebra has no non-zero nil ideals. The question whether the sum of one-sided nil ideals is a one-sided nil ideal is Köthe's problem. It is not known (1982) whether simple nil rings exist. When conditions "of Burnside type" hold in a nil algebra, the algebra is frequently nilpotent or locally nilpotent; a Noetherian nil algebra is nilpotent; so is an Artinian (in particular, a finite-dimensional) nil algebra; a nil algebra of bounded index (the nilpotency indices of the elements are uniformly bounded) over a field of characteristic zero is nilpotent (Higman's theorem); a nil algebra satisfying a polynomial identity is locally nilpotent. It is not clear (1982) whether a finitely-generated nil algebra is nilpotent.

Of special interest are conditions under which the Jacobson radical $\mathrm{Rad}(A)$ of an algebra $A$ over a field $k$ coincides with the nil radical. Here are some of them: $A$ is an Artinian, in particular a finite-dimensional, algebra ($\mathrm{Rad}(A)$ is even nilpotent); $\mathrm{card}(k) > \mathrm{dim}_k A$, in particular, when $A$ is finitely generated over $k$ and $k$ is uncountable; an algebraic algebra $A$; $A$ is finitely generated over $k$ with a polynomial identity, and $k$ is infinite. The radical of a finitely-generated algebra with a polynomial identity over a field of characteristic zero is nilpotent. This is equivalent to the condition that in such an algebra a certain standard identity holds.

Some of the assertions stated above have analogues in non-associative algebras. For example, in an alternative ring with the maximum condition for right ideals and whose additive group contains no elements of orders 2 or 3, every one-sided nil ideal is nilpotent. In a Jordan algebra $A$ over a field $k$ the Jacobson radical $\mathrm{Rad}(A)$ is a nil ideal, provided that one of the following conditions holds: $\mathrm{card}(k) > 2 + \mathrm{dim} A$ or $A$ is an algebraic algebra. Every alternative or special Jordan nil ring of bounded index and without elements of order 2 in the additive group is locally nilpotent (Shirshov's theorem). A finite-dimensional generalized standard nil algebra is nilpotent.

Every anti-commutative algebra is a nil algebra in the sense defined above; therefore, in the class of anti-commutative rings the concept of a nil algebra is meaningless. However, various analogues of this concept turn out to be useful. Thus, in the class of Lie rings an analogue of a nil algebra is an Engel algebra, that is, one in which the inner derivations of elements are nilpotent. An Engel algebra is not necessary locally nilpotent; however, if the nilpotency indices of the inner derivations are uniformly bounded and the ground field has characteristic zero, then an Engel algebra is locally nilpotent. It is not known (1982) whether under these restrictions it is nilpotent (Higgins' problem).

Comments

The Jacobson radical of a polynomial ring $A[x_1,\ldots,x_n]$ in commutative indeterminates $\{x_1,\ldots,x_n\}$ is a polynomial ring $N_n[x]$ over a nil ideal $N_n(A)$ of $A$. Then $N_1 \supseteq N_2 \supseteq \cdots N_\infty = \cap N_n$. $N_\infty$ is characterized as the maximal ideal such that every finitely-generated submodule of nil elements is of bounded index. If $A$ is an algebra over an uncountable field, then all $N_i$ are equal to the upper radical, but generally no other relation is known.

The Köthe problem is equivalent to the following problem: If $A$ is a nil algebra, then is the matrix algebra $M_2(A)$ also nil? A similar problem is the question about algebraic algebras (Kurosh' problem), and the answers are unknown (1989), except in the special cases of $\Pi$-algebras.

References

[1]	N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[2]	I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968)
[3]	A.I. Kostrikin, "Lie rings satisfying the Engel condition" Izv. Akad. Nauk SSSR Ser. Mat. , 21 : 4 (1957) pp. 515–540 (In Russian)
[4]	A.I. Shirshov, "On certain non-associative nil rings and algebraic algebras" Mat. Sb. , 41 : 3 (1957) pp. 381–394 (In Russian)
[5]	E.S. Golod, "On nil algebras and finitely approximable $p$-groups" Izv. Akad. Nauk SSSR Ser. Mat. , 28 : 2 (1964) pp. 273–276 (In Russian)
[6]	G. Higman, "On a conjecture of Nagata" Proc. Cambridge Philos. Soc. , 52 (1956) pp. 1–4
[7]	P.J. Higgins, "Lie rings satisfying the Engel condition" Proc. Cambridge Philos. Soc. , 50 (1954) pp. 8–15
[8]	K.A. Zhevlakov, "The lower nil radical of alternative rings" Algebra i Logika , 6 : 4 (1967) pp. 11–17 (In Russian)
[9]	K. McCrimmon, "The radical of a Jordan algebra" Proc. Nat. Acad. Sci. USA , 612 : 3 (1969) pp. 671–678
[a1]	S.A. Amitsur, "Nil radicals, historical notes" , Coll. Math. Soc. J. Bolyai, Keszthely (Hungary) , 6. Rings and modules and radicals (1971) pp. 47–65