MATHEMATICS

A. I. SHIRSHOV

ON THE LEVITZKI PROBLEM

(Presented by Academician P. S. Aleksandrov, 7 I 1958)

An associative ring \(S\) is called a nil-ring if every element of the ring \(S\) is nilpotent. Levitzki \((^4)\) posed the problem: is every nil-ring nilpotent? This problem was solved affirmatively by Levitzki himself \((^5)\) for the case when the indices of nilpotency of all elements of the ring \(S\) are bounded in the aggregate. Later Kaplansky \((^2)\), dealing with the solution of a more general problem of Kurosch \((^3)\), extended Levitzki’s result to nil-rings with polynomial identities. In the present note an affirmative solution of Levitzki’s problem is given for a broader class of rings introduced for consideration by Dresin \((^1)\).

Let \(\Lambda=\{\lambda_i\}\), \(i=1,2,\ldots,h\), be some set of variables, and let
\(\pi(\lambda)=\lambda_{i_1}\lambda_{i_2}\cdots\lambda_{i_k}\) be some monomial in these variables. Denote by \(T_\pi(\lambda)\) the set of all monomials in the set \(\Lambda\) of degree \(\geq k\) and distinct from \(\pi(\lambda)\).

For any sequence of elements \(\{x_i\}\), \(i=1,2,\ldots,h\), of the ring \(S\), by \(\pi(x)\) we denote the element
\(x_{i_1}x_{i_2}\cdots x_{i_k}\), and by \(T_\pi(x)\) the set of elements of the ring \(S\) obtained by replacing all variables \(\lambda_i\) by the corresponding elements \(x_i\) in each monomial of the set \(T_\pi(\lambda)\).

If there exists a monomial \(\pi(\lambda)\) such that, for any choice of elements \(x_i\), \(i=1,2,\ldots,h\), of the ring \(S\), the element \(\pi(x)\) lies in the right ideal generated by the set \(T_\pi(x)\), then the monomial \(\pi(\lambda)\) is called a strictly supporting monomial of the ring \(S\), and the ring \(S\) a ring with a strictly supporting monomial. For brevity we shall call such rings \(SP\)-rings.

Dresin \((^1)\) showed that the class of \(SP\)-rings contains rings with the minimal condition for right ideals and rings with polynomial identities. In the same work it is shown that, for any \(SP\)-ring, the monomial \(\pi(\lambda)\) may be assumed linear in each of the variables \(\lambda_i\). Under certain strong restrictions Dresin, making use, in essence, of Kaplansky’s methods, gave a positive solution of the Kurosch problem for \(SP\)-algebras, i.e. proved the local finiteness of algebraic \(SP\)-algebras of a certain special kind. However, Dresin himself notes the difficulties which did not allow him to solve even Levitzki’s problem for \(SP\)-rings without additional restrictions.

If the strictly supporting monomial \(\pi(\lambda)\), which in what follows we shall assume to be linear in each variable \(\lambda_i\), has degree \(t\), then the \(SP\)-ring \(S\) will be called an \(SP\)-ring of degree \(t\).

Lemma. Let \(S\) be a nil-\(SP\)-ring of degree \(t\), and let \(I\) be its ideal generated by the elements \(a_i^t\), where \(a_i\), \(i=1,2,\ldots,n\), is some fixed set of elements of the ring \(S\).

Then for every natural number \(q>t\) there exists a natural number \(k=k(q)\) such that the ideal \(I^k\) belongs to the ideal generated by the elements \(a_i^q\).

Proof. Suppose there exists a natural number \(r\) such that the ideal \(I^r\) lies in the ideal generated by the elements \(a_i^m\), \(i=1,2,\ldots,n\); \(m \ge t\). To prove the lemma we shall show that there exists a natural number \(r_1\) such that the ideal \(I^{r_1}\) lies in the ideal generated by the elements \(a_i^{m+1}\).

Every element of the ideal \(I^{r(nt+1)}\) can be represented as a sum of products of \(nt+1\) elements of the form \(\alpha a_i^m \beta\), where \(\alpha\) and \(\beta\) are monomials in the generators of the ring \(S\). For each such product there is an element \(a_j^m\) occurring in it at least \(t+1\) times. Thus each such product can be written in the form
\[ c_1Dc_{t+2}=c_1d_1d_2\ldots d_tc_{t+2} =c_1(a_j^m c_2a_j)a_j^{m-1}c_3a_j^2)(a_j^{m-2}c_4a_j^3)\ldots(a_j^{m-t+1}c_{t+1}a_j^t)c_{t+2}. \]
By assumption, the monomial \(D\) belongs to the right ideal generated by all products of its factors, different from \(D\) and of degree not less than that of \(D\) with respect to the elements \(d_i\).

For any other monomial of degree \(t\) in the elements \(d_i\) there are two adjacent elements \(d_{j_1}\) and \(d_{j_2}\) such that \(j_1 \ge j_2\). In each such case, in the corresponding segment there will stand the word
\[ d_{j_1}d_{j_2} = a_j^{m-j_1+1}c_{j_1+1}a_j^{j_1}a_j^{m-j_2+1}c_{j_2+1}a_j^{j_2} = a_j^{m-j_1+1}c_{j_1+1}a_j^{m+1+(j_1-j_2)}c_{j_2+1}a_j^{j_2}. \]
It is easy to see that all such elements lie in the ideal generated by the element \(a_j^{m+1}\).

It follows that \(c_1D=\omega_1+c_1Dq\), where \(\omega_1\) is an element of the ideal generated by the element \(a_j^{m+1}\). But then
\[ c_1D=\omega_1+\omega_1q+c_1Dq^2 =\omega_1+\omega_1q+\omega_1q^2+c_1Dq^3=\cdots =\omega_1+\omega_1q+\omega_1q^2+\cdots+\omega_1q^l+c_1Dq^{l+1} \]
for any \(l\). The assertion is proved by virtue of the nilpotency of the element \(q\). As the number \(r_1\) one may take \(r(nt+1)\).

Theorem. Every nil-\(SP\)-ring is locally nilpotent.

Proof. Let \(S\) be a nil-\(SP\)-ring of degree \(t\) with a finite number of generators, and let \(J\) be the ideal generated in it by all possible elements of the form \(a^t\), \(a \in S\). The factor ring \(S/J\) is nilpotent by Levitzki’s theorem \((^5)\), and this means that there exists a natural number \(M\) such that every element of the form \(b_{i_1}b_{i_2}\ldots b_{i_M}\), where the \(b_{i_s}\) are generators of the ring \(S\), belongs to the ideal \(J\). Since there are only finitely many elements of the form \(b_{i_1}b_{i_2}\ldots b_{i_M}\), the ideal \(S^M\) belongs to some ideal \(J_1\), contained in the ideal \(J\) and generated by some finite set of elements of the form \(a_i^t\).

From the lemma it follows that the ideal \(J_1\), and consequently also the ring \(S\), is nilpotent.

Moscow State University
named after M. V. Lomonosov

Received
3 I 1958

REFERENCES

1. M. P. Drazin, Proc. Am. Math. Soc., 8, no. 2, 352 (1957).
2. J. Kaplansky, Trans. Am. Math. Soc., 68, 62 (1950).
3. A. G. Kurosh, Izv. Akad. Nauk SSSR, Ser. Mat., 5, 233 (1941).
4. J. Levitzki, Bull. Am. Math. Soc., 49, 913 (1943).
5. L. Levitzki, Bull. Am. Math. Soc., 52, 1033 (1946).