Journal of Algebra 324 (2010) 3044–3047

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Journal of Algebra
www.elsevier.com/locate/jalgebra

Nil subrings of endomorphism rings of finitely generated

modules over affine PI-rings ✩
Robert M. Guralnick a,∗ , Lance W. Small b , Efim Zelmanov b
a
Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA
b
Department of Mathematics, University of California, San Diego, La Jolla, CA 92093 USA

a r t i c l e i n f o a b s t r a c t

Article history: We prove that nil subrings of certain endomorphism rings are
Received 7 October 2009 nilpotent and obtain some nilpotence results about Lie subrings.
Available online 23 June 2010 © 2010 Elsevier Inc. All rights reserved.
Communicated by Nicolás Andruskiewitsch

Dedicated to Susan Montgomery on her

65th birthday

MSC:
primary 16R10
secondary 16U80

Keywords:
Nil subrings
Affine PI-rings
Endomorphism rings

1. Introduction

Since Levitzki [L], there has been interest in showing that nil subrings of endomorphisms of finitely
generated modules over various classes of rings are nilpotent. Hunter [H] showed, for example, that
such results can be applied to show that various weakly closed nil subsets generate associative nilpo-
tent subrings – generalizing, for instance, Engel’s theorem.
Let R be an affine PI-algebra over a commutative noetherian ring. In this note, we show that nil
subrings of the endomorphism ring of finitely generated modules over R are nilpotent. This result

✩
The authors gratefully acknowledge the support of the NSF (grants DMS-0653873 and DMS-0758487).
*Corresponding author.
E-mail addresses: guralnic@usc.edu (R.M. Guralnick), lwsmall@ucsd.edu (L.W. Small), ezelmano@math.ucsd.edu
(E. Zelmanov).

0021-8693/$ – see front matter © 2010 Elsevier Inc. All rights reserved.
doi:10.1016/j.jalgebra.2010.06.005
 R.M. Guralnick et al. / Journal of Algebra 324 (2010) 3044–3047 3045

extends the famous theorem of Braun [B] that nil subrings of affine PI-rings are nilpotent (we do use
Braun’s result in our proof). For background and basic PI theory we refer the reader to Rowen [R].
Our main result is:

Theorem 1.1. Let R be an affine-PI algebra over the commutative noetherian ring C , M R a finitely generated
R-module and S = End R ( M ). If V is a nil subring of S, then V is nilpotent.

Note that it is not true that S need be an affine PI-ring. An easy example of this is to let k be a
field, W = k[x, y ] and
 
k + yW W
R= .
yW W

Let M = e 11 R. Then one easily sees that S = e 11 Re 11 is not noetherian (and so not affine over any
noetherian subring).
Combining this result with the main result of Hunter [H], we immediately obtain:

Corollary 1.2. Let R be an affine PI algebra over the commutative noetherian ring C , M R a finitely generated
R-module and S = End R ( M ). If V is a nil subset of S closed under the Lie bracket, then V generates a nilpotent
ring.

In fact, the previous result holds under the more general assumption that V is “weakly closed”.
Recall that a Lie ring is said to be weakly Engel, if given x ∈ L, there is a positive integer n = n(x)
such that ad(x)n(x) ( y ) = 0 for all y ∈ L. We show that the previous corollary implies:

Corollary 1.3. Let A be an affine PI-algebra over a commutative noetherian ring C . If L is a weakly Engel Lie
subring of A, then L is a nilpotent Lie ring.

We should point out that we do not obtain the Jacobson property for endomorphism rings of
finitely generated modules over affine PI-rings.
In the next section, we show that one can reduce to the case that M R is a cyclic module and that
R is prime. In the next two sections we prove the main result and the corollary and give some other
easy corollaries about Lie subrings of affine PI-rings.

2. Preliminary results and reductions

Let R be an affine PI algebra over a commutative noetherian ring C .

We next recall two well-known facts.

(i) ([P]) R satisfies the ascending chain condition on prime ideals.

(ii) (Braun’s theorem [B]) There exist prime ideals P 1 , . . . , P t for some positive integer t with
P 1 P 2 · · · P t = (0).

Assume that we have proved the theorem if R is prime and M is cyclic. We want to show that the
theorem follows.
We assume the hypotheses of the theorem. Replacing M by n copies of M and R by M n ( R ) for n
sufficiently large allows us to assume by standard Morita theory that M R is cyclic.
Let M R ∼ = R / I be a cyclic R-module for some right ideal I of R. Thus, End R ( R / I ) = Π( I )/ I , where
Π( I ) is the idealizer of I in R. By Braun’s theorem, P 1 P 2 · · · P t = (0) for some prime ideals P 1 , . . . , P t .
Suppose that V is a subring of Π( I ) containing I with V / I nil. Since we are assuming the result
for prime rings, we see that there exist positive integers ni such that V ni ⊆ P i + I . Since I ⊆ V , this
implies that V ni ⊆ I + ( P i ∩ V ). Thus, V n1 V n2 · · · V nt ⊂ ( I + ( P 1 ∩ V ))( I + ( P 2 ∩ V )) · · · ( I + ( P t ∩ V )) ⊆ I
as was to be shown.
3046 R.M. Guralnick et al. / Journal of Algebra 324 (2010) 3044–3047

3. Prime rings

In this section, we give the proof of the theorem when R is prime and M R is cyclic. As we noted
in the previous section, this is sufficient to prove the theorem as stated.
Suppose the result is false. Let P be a maximal element in the set of prime ideals of R such that
the theorem fails for R / P (this exists since R has ACC on prime ideals). By passing to R / P , we may
assume that every proper prime quotient of R does satisfy the theorem.
It follows that R / X satisfies the theorem for every non-zero ideal X of R (since it holds for R / Q
for every prime ideal Q containing X and so as in the previous section, it then holds for R / X ).
Set M = R / I and End R ( M ) = Π( I )/ I where Π( I ) = {r ∈ R | r I ⊆ I } is the idealizer of I . Further, let
N ⊇ I be a subring of Π( I ) such that for each n ∈ N, there is a positive integer s = s(n) such that
ns ∈ I . We need to show that N t ⊆ I for some positive integer t.
By Posner’s theorem, R has a classical ring of quotients Q with Q a finite-dimensional simple
ring. Additionally, Q = Z −1 R where Z is the center of R (cf. [R, 1.7.3]). Thus, I Q = e Q where e is an
idempotent in Q and e = iz−1 for some z ∈ Z and i ∈ I . Since R / zR satisfies the result, N m ⊆ I + zR
for some positive integer m. Since N is contained in the idealizer of I Q in Q , it follows by Levitzki’s
theorem [L] that N n ⊆ I Q = e Q for some positive integer n. Thus, eN n = N n . So N n zR = eN n zR =
iN n R ⊆ I . Hence N m+n ⊆ N n ( I + zR ) ⊆ I as required.

4. Lie rings

We now prove Corollary 1.3.

View A as a left A ⊗C A op := R module. Identify L with

L̂ := { ⊗ 1 − 1 ⊗  |  ∈ L } ⊂ A ⊗C A op .

Then L acts on L ⊆ A. Let ( L ) be the left annihilator of A in R. By hypothesis, ad(x)n(x) ∈ ( L ) for all
x ∈ L or equivalently, some power of each element of L̂ is in ( L ). By a result of Regev (see [R, 6.1.4]),
R is an affine PI-ring over C .
Note that ( L )ad(x) ⊆ ( L ) (because if y ∈ ( L ) and  ∈ L, then y (x ⊗ 1 − 1 ⊗ x) = y [x, ] = 0).
Thus, L̂ is in the idealizer X of ( L ). So L̂ is a nil Lie subring of X /( L ) and thus by Corollary 1.2,
it generates a nilpotent subring. In particular, this implies that L̂ N ⊆ ( L ) for some N > 0. Note that
L̂ ∩ ( L ) = Z ( L̂ ) and so L / Z ( L ) is a nilpotent Lie algebra, whence L is also nilpotent.
We also obtain as an immediate corollary of Theorem 1.1 the following generalization of results in
[RW,S].

Corollary 4.1. Let R be an affine PI-ring over a noetherian commutative ring C . If R is a weakly Engel Lie
algebra, then R is nilpotent as a Lie algebra.

In both [RW,S], the authors assumed that R was actually Engel rather than weakly Engel – in
particular, R is a PI-algebra.
We point out one more consequence in this setting.

Corollary 4.2. Let R be an affine PI-ring over a commutative noetherian ring C . If R is a weakly Engel Lie
algebra, then [ R , R ] generates a nilpotent ideal of R.

Proof. By Braun’s theorem, we may assume that R is semiprime and then prime. By the previous
result, R is a nilpotent Lie algebra, whence also the quotient ring Q of R is nilpotent as a Lie algebra.
Since Q ∼ = Mn ( D ) for some finite-dimensional division algebra, it follows that n = 1. We claim that
D is a field. It suffices to assume that D is infinite, whence we can extend scalars and reduce to the
matrix case. Thus, [ R , R ] = 0 and the result follows. 2
 R.M. Guralnick et al. / Journal of Algebra 324 (2010) 3044–3047 3047

References

[B] A. Braun, The nilpotency of the radical of a finitely generated PI-ring, J. Algebra 89 (1984) 375–396.
[H] K. Hunter, Nilpotence of nil subrings implies more generalized nilpotence, Acta Math. (Basel) 18 (1967) 136–139.
[L] J. Levitzki, Über nilpotente Unterringe, Math. Ann. 105 (1931) 620–627.
[P] C. Procesi, Non-commutative affine rings, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I 8 (1967) 237–255.
[RW] D. Riley, M. Wilson, Associative rings satisfying the Engel condition, Proc. Amer. Math. Soc. 127 (1999) 973–976.
[R] L.H. Rowen, Polynomial Identities in Ring Theory, Academic Press, New York, 1980.
[S] A. Shalev, On associative algebras satisfying the Engel condition, Israel J. Math. 67 (1989) 287–290.