«Abstract. In this paper we survey some results on the structure of noncommutative rings. We focus particularly on nil rings, Jacobson radical rings ...»
Some results in noncommutative ring theory
Abstract. In this paper we survey some results on the structure of noncommutative rings. We
focus particularly on nil rings, Jacobson radical rings and rings with ﬁnite Gelfand–Kirillov
Mathematics Subject Classiﬁcation (2000). 16-02, 16-06, 16N40, 16N20, 16N60, 16D60,
Keywords. Nil rings, Jacobson radical, algebraic algebras, prime algebras, growth of algebras,
the Gelfand–Kirillov dimension.
1. Introduction We present here a brief outline of results and examples related mainly to noncommu- tative nil rings. In this exposition rings are noncommutative and associative. A vector space R is called an algebra (or a K-algebra) if R is equipped with a binary operation ∗ : (R, R) → R, called multiplication, such that for any a, b, c ∈ R and for any α ∈ K, we have (a + b) ∗ c = a ∗ c + b ∗ c, a ∗ (b + c) = a ∗ b + a ∗ c, (a ∗ b) ∗ c = a ∗ (b ∗ c), α(a ∗ b) = (αa) ∗ b = a ∗ (αb).
It is known that simple artinian rings, commutative simple rings and simple right noetherian rings of characteristic zero have unity elements . In this text, rings are usually without 1. In fact nil rings and Jacobson radical rings cannot have unity elements.
2. Nil rings The most important question in this area is the Köthe Conjecture, ﬁrst posed in 1930.
Köthe conjectured that a ring R with no nonzero nil (two-sided) ideals would also have no nonzero nil one-sided ideals, , see also  and . This conjecture is still open despite the attention of many noncommutative algebraists. It is a basic question concerning the structure of rings.
Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society 260 Agata Smoktunowicz The truth of the conjecture has been established for many classes of rings: typically, one proves that for a given class of rings, the sum of all nil one-sided ideals is nil. The most famous examples of such results are the proof of the conjecture in the case of algebras over uncountable ﬁelds by Amitsur, and the fact that nil ideals are nilpotent in the class of noetherian rings, proved by Levitski, see . However, as indicated above, Köthe’s conjecture is still open in the general case.
An element r in a ring R is said to be nilpotent if r n = 0 for some n. A ring R is a nil ring if every element of R is nilpotent, and the ring R is nilpotent if R n = 0 for some n. A more appropriate deﬁnition in the case of inﬁnitely generated rings is the following. A ring R is locally nilpotent if every ﬁnitely generated subring of R is nilpotent. A thoroughunderstanding of nil and nilpotent rings is important for an attempt to understanding general rings.
In addition, nil rings have some applications in group theory. The following famous theorem was proved in 1964 by Golod and Shafarevich. For every ﬁeld F there exists a ﬁnitely generated nil F -algebra R which is not nilpotent (). Recall that a group G is said to be torsion (or periodic) if every g ∈ G has a ﬁnite order. Golod used the group 1 + R, when F has positive characteristic, to get a counterexample to the General Burnside Problem: Let G be a ﬁnitely generated torsion group. Is G necessarily ﬁnite?
There are many open questions concerning nil rings. As mentioned before, the most important is now known as the Köthe Conjecture and was posed by Köthe in 1930: if a ring R has no nonzero nil ideals, does it follow that R has no nonzero nil one-sided ideals? Köthe himself conjectured that the answer would be in the afﬁrmative (, , ).
There are many assertions equivalent to the Köthe Conjecture: For example, the
following are equivalent to Köthe’s conjecture:
1. The sum of two right nil ideals in any ring is nil.
2. (Krempa ) For every nil ring R the ring of 2 by 2 matrices over R is nil.
3. (Fisher, Krempa ) For every ring R, R G is nil implies R is nil (G is the group of automorphisms of R, R G the set of G-ﬁxed elements).
4. (Ferrero, Puczylowski ) Every ring which is a sum of a nilpotent subring and a nil subring must be nil.
5. (Krempa ) For every nil ring R the polynomial ring R[x] in one indeterminate over R is Jacobson radical.
6. (Smoktunowicz ) For every nil ring R the polynomial ring R[x] in one indeterminate over R is not left primitive.
Recall that a ring R is Jacobson radical if for every r ∈ R there is r ∈ R such that r + r + rr = 0. Every nil ring is Jacobson radical. The largest ideal in a ring R, which is Jacobson radical is called the Jacobson radical of R. The Jacobson radical of a ring R equals the intersection of all (right) primitive ideals of R (I is a primitive ideal in R if I /R is primitive). Recall that a ring R is (right) primitive if there is a maximal right ideal Q such that Q + I = R for every nonzero ideal I in R and there is b ∈ R such that br − r ∈ Q for every r ∈ R ().
The Köthe Conjecture is said to hold for a ring R if the ideal generated by the nil left ideals of R is nil. Köthe’s conjecture holds for the class of Noetherian rings (Levitzki, , ), Goldie rings (Levitzki, ), rings with right Krull dimension (Lenagan , ), monomial algebras (Beidar, Fong ), PI rings (Rasmyslow– Kemer–Braun , , ,  ), algebras over uncountable ﬁelds (Amitsur , ).
There are many related results, some are indicated in the following.
Theorem 2.1 (Levitzki; ).
Let R be a right Noetherian ring. Then every nil one-sided ideal of R is nilpotent.
Theorem 2.2 (Lenagan ).
If R has right Krull dimension, then nil subrings of R are nilpotent.
Theorem 2.3 (Gordon, Lenagan and Robson, Gordon and Robson; ).
If R has right Krull dimension, then the prime radical of R is nilpotent.
The prime radical of R is a nil ideal and is equal to the intersection of all prime ideals in R.
Theorem 2.4 (Beidar, Fong ).
Let X be a nonempty set, Z = X the free monoid on X, Y an ideal of the monoid Z, and F a ﬁeld. Then the Jacobson radical of the monomial algebra F [Z/Y ] is locally nilpotent.
In the case of characteristic zero the result is due to Jaspers and Puczylowski, . Earlier, Belov and Gateva-Ivanova  showed that the Jacobson radical of a ﬁnitely generated monomial algebra over a ﬁeld is nil. However, it is not true that the Jacobson radical of a ﬁnitely generated monomial algebra is nilpotent, since it was shown by Zelmanov  that there is a ﬁnitely generated prime monomial algebra with a nonzero locally nilpotent ideal.
Theorem 2.5 (Razmyslov–Kemer–Braun , , ; ).
If R is a ﬁnitely generated PI-algebra over a ﬁeld then the Jacobson radical of R is nilpotent.
Razmyslov  proved this for rings satisfying all identities of matrices, Kemer  for algebras over ﬁelds of characteristic zero. Later Braun  proved the nilpotency of the radical in any ﬁnitely generated PI algebra over a commutative noetherian ring. Amitsur has previously shown that the Jacobson radical of a ﬁnitely generated PI algebra over a ﬁeld is nil.
Another famous result is the Nagata–Higman Theorem:
262 Agata Smoktunowicz Theorem 2.6 (Nagata–Higman; ). If A is an associative algebra of characteristic p such that a n = 0 for all a ∈ A and p n or p = 0 then A is nilpotent.
For interesting results related to Nagata–Higman’s theorem see .
A theorem of Klein  asserts that if R is a nil ring of bounded index then R[x] is a nil ring of bounded index.
In 1956 Amitsur  showed that if R is a nil algebra over an uncountable ﬁeld, then the polynomial ring R[x] in one indeterminate over R is also nil. The situation is completely different for countable ﬁelds, as was shown by the author in 2000.
Theorem 2.7 (Smoktunowicz ).
For every countable ﬁeld K there is a nil Kalgebra N such that the polynomial ring in one indeterminate over N is not nil.
This answers a question of Amitsur. Another important theorem by Amitsur is the following.
Theorem 2.8 (Amitsur; ).
Let R be a ring. Then the Jacobson radical of the polynomial ring R[x] is equal to N[x] for some nil ideal N of R.
In 1956 Amitsur conjectured that if R is a ring, and R[x] has no nil ideals then it is semiprimitive (i.e. the Jacobson radical of R[x] is zero). This assertion is true for many important classes of rings, as mentioned above. However, the following theorem shows that this conjecture does not hold in general: There is a nil ring N such that the polynomial ring in one indeterminate over N is Jacobson radical but not nil (). For some generalizations of this theorem see . This theorem is true in a more general setting: For every natural number n, there is a nil ring N such that the polynomial ring in n commuting indeterminates over N is Jacobson radical but not nil.
Recall that, as shown by Krempa in , Köthe’s conjecture is equivalent to the assertion that polynomial rings over nil rings are Jacobson radical. However, homomorphic images of polynomial rings over nil rings with nonzero kernels are often Jacobson radical, as is shown by the next result.
Theorem 2.9 (Smoktunowicz ).
Let R be a nil ring and R[x] the polynomial ring in one indeterminate over R. Let I be an ideal in R[x] and M the ideal of R generated by coefﬁcients of polynomials from I. Then R[x]/I is Jacobson radical if and only if R[x]/M[x] is Jacobson radical.
The following are interesting open questions on nil rings.
Question 1 (Latyshev, , pp. 12). Let A be an associative algebra with a ﬁnite number of generators and relations. If A is a nil algebra must it be nilpotent?
3. Algebraic algebras The most well-known question in this area is the Kurosh Problem (, ). Let R be a ﬁnitely generated algebra over a ﬁeld F such that R is algebraic over F. Is R ﬁnite dimensional over F ?
This problem has a negative solution in general. The famous construction of Golod and Shafarevich in the 1960s produced a ﬁnitely generated nil algebra which is not nilpotent (). This was then used to construct a counterexample to the Burnside Conjecture, one of the biggest outstanding problems in group theory at that time. Zelmanov was later awarded the Fields Medal for his solution of the Restricted Burnside Problem .
However, the Kurosh Problem is still open for the key special case of a division
Question 3 (Kurosh’s problem for division rings , ). Let R be a ﬁnitely generated algebra over a ﬁeld F such that R is algebraic over F and R is a division ring. Does it follow that R a ﬁnite dimensional vector space over its center?
Again, as with the nil ring problems, there are many partial results. The Kurosh Problem for division rings is still open in general, but it is answered afﬁrmatively for F ﬁnite and for F having only ﬁnite algebraic ﬁeld extensions, in particular, for F algebraically closed (). By Levitzki’s and Kaplanski’s theorem, Kurosh’s conjecture is also true if there is a bound on the degree of elements in R ().
It is unknown whether Kurosh’s problem for division rings has a positive answer in the case of algebras over uncountable ﬁelds. Also the following question is still open: Is Kurosh’s conjecture true for division rings with ﬁnite Gelfand–Kirillov dimension, and in particular for division rings with quadratic growth? There are obvious connections with problems in nil rings. A nil element is obviously algebraic, and, in the converse direction, it is possible to construct an associated graded algebra connected with an algebraic algebra in such a way that the positive part is graded nil, i.e., all homogeneous elements are nil. On the other hand, the Kurosh Problem has a negative solution for rings with ﬁnite Gelfand–Kirillov dimension (), for simple rings (), for primitive rings (), for ﬁnitely generated primitive rings (), and for ﬁnitely generated algebraic primitive rings (). However, a natural question
arising from the general Kurosh Problem remains open:
Question 4 (Small’s question). Let R be a ﬁnitely generated simple algebra with 1 over a ﬁeld F such that R is algebraic over F. Is R a ﬁnite dimensional vector space over its center?
Another open question on division rings, which has been around for years, is the
Question 5. Let K be a ﬁeld and let R be a ﬁnitely generated algebra which is a division ring. Does it follow that R is a ﬁnitely generated vector space over K?
264 Agata Smoktunowicz As far as I know this question is very much open even with various conditions, like e.g. Gelfand–Kirillov dimension 2. It has been shown by Small () that a division ring which is a homomorphic image of a graded noetherian ring (of course, by a non graded ideal) must be ﬁnite dimensional. There is a similar open question
Question 6 (, p. 20). Does there exist an inﬁnite associative division ring which is ﬁnitely generated as a ring?