The Lie structure of a commutative ring with a derivation

9 1985 Birkh/iuser Verlag, Basel

The Lie structure of a commutative ring with a derivation

By ANDRZEJ NOWICKI

Introduction. Let R be a commutative ring with identity and let d be a non-zero derivation of R. By R o we denote the Lie ring whose elements are those of R and whose product is as follows:

[a, b] = ad(b) - d(a) b, for all a, b 9 R (see [4], [14]).

By {R,} we denote the sequence R o ~ R 1 ~ ... of Lie subrings of R o defined by R. + 1 = [R,, Ro], for all n > 0. Let A be a non-zero Lie ideal of R o and assume that R is 2-torsion free.

In this situation C.R. Jordan and D.A. Jordan ([3], [4]) proved the following four theorems:

(1) I f R is prime and A = R o or A = R 1 then A is a prime Lie ring ([3] Theorem 6, [4]

Theorems I and 4).

(2) If R is noetherian d-prime and A = R o or A = R 1 then A is a prime Lie ring ([3]

Theorem 7, [4] Theorems 2 and 5).

(3) If R is noetherian d-prime and A = R o or A = R 1 then every non-zero Lie ideal of A contains a non-zero d-ideal of R ([3] Theorem 8, [4] Theorems 3 and 6).

[4] If R is noetherian then R o is simple if and only if R is d-simple ([4] Theorem 3).

In this paper we show that Theorems (1)-(3) are also true in the case where A = R,, for any n > 0 (Section 4), and that (4) also holds for non-noetherian rings (Theorem 3.3).

Moreover we show, with the additional assumption (namely 6 is not a zero-divisor in R), that Theorems (1)-(3) are true for every Lie ideal A of R o (Corollary 2.7, Corollary 3.5).

We obtain these results as consequences of more general facts which we prove in this paper (see for example Theorems 2.5 and 3.4).

1. Preliminaries. Throughout the paper R will be a commutative ring with identity and d a non-zero derivation of R, N will stand for the nil radical, and 3 (R) for the set of zero divisors of R.

By R o we denote the Lie ring whose elements are those of R and whose product is as follows:

[a, b] = ad(b) - d(a) b, for all a, b 9 R (see [4], [14]).

Vol. 45, 1985 Lie structure of a ring with a derivation 329 If A and B are subsets of R we denote by A B the additive subgroup of R, generated by all elements of R of the form a b, a e A, b e B, and by [A, B] the additive subgroup of Ro, generated by all elements of R o of the form [a, b], a e A, b e B.

If X is a subset of R we put X~ for the set {r e R: d"(r) ~ X for n = 0, 1,...}.

An ideal I of the ring R is said to be a d-ideal of R if d(I) c= I.

It is easy to prove the following two lemmas.

Lemma 1.1. Let I be an ideal of R. Then (a) I ~ is the largest d-ideal contained in I, (b) ~ d"(I) is the smallest d-ideal containing I,

n = O k k + l

(c) If R is noetherian then there exists an integer k such that ~ d"(I) = ~ d"(1) is a

d-ideal of R. . = o . = o

Lemma 1.2. Let I be an additive subgroup of R.

(a) If I is a Lie ideal of R o then d(I) ~= I.

(b) I is a d-ideal of R if and only if I is both an ideal of R and a Lie ideal of R o.

A Lie ring L is said to be prime if the product of non-zero Lie ideals of L is always non-zero and to be simple if L has no Lie ideals other than 0 and L (see [2], [7]).

The ring R is called d-prime if the product of non-zero d-ideals of R is always non-zero and is called d-simple if R has no d-ideals other than 0 and R.

The ring R is called quasi-prime ([8] -[10], [11], [12]) if there exists a multiplicative subset S of R such that 0 is maximal among d-ideals of R disjoint from S. We see at once that if R is domain or R is d-simple then R is quasi-prime.

Quasi-prime rings may be characterized in several useful ways.

Proposition 1.3. The following conditions are equivalent:

(1) R is quasi-prime,

(2) There exists a prime ideal P of R such that P~ = 0, (3) N is a prime ideal of R and Ng = O,

(4) 0 is a primary ideal of R and N~ = O, (5) N = 3 ( R ) andN e =0.

P r o of. The equivalence of conditions (1)-(4) is given in [9] (Proposition 2.2). The equivalence of (4) and (5) is obvious.

We note also the following two facts.

Proposition 1.4.

If R is quasi-prime then R is d-prime.

P r o of. See [12] Proposition 2.1. The converse is not necessarily true (see [12]).

Proposition 1.5. Let R be noetherian. Then R is quasi-prime if and only if R is d-prime.

P r o o f . See [5] or [12] Theorem 2.1.

2. Normal sequences of prime Lie subrings ofR o. A sequence R o = A o ~ A i ___ ^{A 2 ~ . . .} of Lie subrings of R o is said to be normal (see [1]) if A n + a is a Lie ideal of A , , for all n _-> 0.

Let R o = A o ~ A1 ~ A2 ~ . . . be a normal sequence of Lie subrings of the Lie ring R o.

The aim of this section is to show that if R is quasi-prime, n is a fixed natural number and (n + 2)! ~ 3 (R) then A n is prime Lie ring.

Lemma 2.1. Let R be quasi-prime. If I is a Lie ideal of R o such that I ~= N then I = O.

P r o o f . Let / ~ N. Then, by Lemma 1.2(a), d"(/) ~ N for any natural n. Hence / ~ N e . But by Proposition 1.3, N~ = 0 and hence I = 0.

Lemma 2.2. Let R be quasi-prime, {Ai} be a normal sequence of Lie subrings of R o and n be a natural number. If I is a Lie ideal of the ring A n such that I c= N then I = O.

P r o o f (by induction on n). Lemma 2.1 gives us the case n = 0. Let n > 0 and assume that the lemma is true for all natural numbers < n. Let / be a Lie ideal of A n such that I~=N.

If A n = N then, by induction, A n = 0 (since A n is a Lie ideal of A n a) and hence I = 0.

Assume that A n ~ N.

We show that dk(I) c= N for all k > 0. Since I ___ N then d~ ~= N. Let k > 0 and suppose that dJ(/) ~= N for all j, 0 =<j < k. Let a ~/, x ~ An\N. Then

ad(x) - d(a) x = [a, x] e I, dk- l(ad(x) - d(a) x) ~ N,

SO

that is,

~. dJ(a) dk_J(x)_ ~-, k 1 dJ+l(a)dk_i_J(x)~g"

j=o J j=o

Since a, d(a) .... , dk- i(a) ~ N, it follows that dk(a) x ~ N. Hence, by Proposition 1.3, dk(a) ~ N. Therefore dk(I) ~= N for all k > 0, that is, 1 ~ Ne. But by Proposition 1.3, Ne = 0 and hence I = 0. The result follows by induction.

If r, a 1 , . . . , a, are elements of R then by (r, al . . . . , a , ) we denote the element [[... [[r, al], a2] . . . . ], a,]. For example,

(r, a,) = -- ald(r ) + d(ai) r,

(r, ai, a2) = ai a2d2(r) - aid(a2) d(r) + d(d(ai) a2) r, and, by induction, we obtain

Lemma 2.3. If r, a i . . . a, ~ R then

(r, ai . . . a , ) = ( - l)"al ' " a,d"(r) + u,_id" i(r) + . ' - + uid(r ) + Uo r, where Uo, ..., u n i are elements in R independent from r.

Vol. 45, 1985 Lie structure of a ring with a derivation 331

Note also the following lemma proved in [13].

Lemma2.4.

Let f = und" + u n_ i d"- l + . . . + ui d + u od o , where u o .... , u, e R. If f(R) = 0 then

n! und(r)" = 0 for all r ~ R.

Now we are able to prove the main result of this section.

Theorem

2.5. Let R be quasi-prime and let R o = A o ~= A i ~= ... be a normal sequence of non-zero Lie subrings of R o. If, for some natural n, (n + 2)! ~ 3(R) then A n is a prime Lie ring.

P r o o f . Let I, J be Lie ideals of A, such that [I, J] = 0. We show that I = 0 or J = 0 . Let r e R , a l e A 1 . . . a , ~ A , , i e I and j ~ J . Then (r, a i , . . . , a , , i ) eI, so (r, a i . . . a,, i,j> = 0. By Lemmas 2.3 and 2.4 it follows that

(n + 2)! a i a 2 . . , anijd(r) "+2 = O, that is,

a i a 2 . . , a, ijd(r) n+2 = O.

Observe that, by Lemma 2.1, d(R) ~ N (since d 4= 0 and Rd(R) is a Lie ideal of Ro). Hence, by Proposition 1.3 and Lemma 2.2, ij = 0 for all i e I, j e J. Therefore if I ~ N then, by Lemma 2.2, I = 0; and i f / ~ : N then, by Proposition 1.3, J = 0. This completes the proof.

As an immediate consequence of Theorem 2.5 we obtain the following generalizations of Theorems 1, 2, 4 and 5 in [4] (see also [3]).

Corollary

2.6. Let 2 ~ 3(R). If R is quasi-prime then R o is a prime Lie ring.

Corollary

2.7. Let 6 d~ 3(R). If R is quasi-prime then any Lie ideal of R o is a prime Lie ring~

3. Simple Lie ring and d-simple differential rings.

C. R. Jordan and D. A. Jordan ([3], [4]) showed that if R is noetherian and 2-torsion-free then Ro is simple if and only if R is d-simple. In this section we prove that this is also true for R non-noetherian (comp. [6]).

Note that obviously there exist non-noetherian d-simple rings (see for example [15]).

We start with the following

Lemma 3.1.

Let

2 ~ 3(R).

If A is a Lie ideal of R o such that

[A, A] =~ 0,

then A contains a non-zero d-ideal of R.

P r o o f. Let a, b be such elements in A that [a, b] + 0. Let x = 2 [a, b] and I = R x.

Observe that I ~ A. In fact, if r e R then rx = 2r[a, b] = [ra, b] + [a, rb] ~ A.

It follows, that I is a non-zero ideal of R contained in A. Furthermore, since d(A) ~ A (by Lemma 1.2), it follows that ~, d"(I) is a non-zero d-ideal of R, which is contained in A.

n = 0

As an immediate consequence of Theorem 2.5 and Lemma 3.1 we have:

Proposition

3.2. Let 2 ~ 3(R). If R is quasi-prime then every non-zero Lie ideal of R o contains a non-zero d-ideal of R.

Now we are ready to prove the following

Theorem 3.3. Let 2 ~ 3(R). Then R o is simple if and only if R is d-simple.

P r o o f. If R is d-simple then R is quasi-prime and, by Proposition 3.2, R 0 is simple.

Now let R o be simple. If I is a non-zero d-ideal of R then I is a non-zero Lie ideal of R 0 and hence I = R. It follows that R is d-simple. This completes the proof.

The following theorem is a generalization of Proposition 3.2 for the case where R is noetherian d-prime. In its proof we use an idea from [4] Theorem 6.

Theorem 3.4. Let R be noetherian d-prime, R o = A o ~= A 1 ~ A z ~= ... be a normal se- quence of Lie subrings of Ro, and n be a natural number. If (n + 2)! d~ 3(R) then every non-zero Lie ideal of the Lie ring A, contains a non-zero d-ideal of R.

P r o o f (by induction on n). If n = 0 then this follows by Propositions 1.5 and 3.2. Let n > 0 and assume that the theorem is true for A , _ 1. Consider the Lie ring A , . The case A , = 0 is trivial. Let A , :~ 0 and let I be a non-zero Lie ideal of A,. Then, by Theorem 2.5, [I, I] + 0. Choose a, b ~ I such that [a, b] :~ 0. Since A , is a non-zero Lie ideal of A , _ 1 there exists, by induction, a non-zero d-ideal U of R contained in A,. Put J = 2 9 [a, b] 9 U.

It is clear that J is an ideal of R. Observe that J ~ I. In fact, if u e U then au, bu ~ A,, hence

2 [a, b] u = [a, b u] + [a u, b] e I.

Observe also that J 4 0. In fact, suppose that J = 0. Then 0 :~ 2[a,b]~Ann(U), i.e.

Ann(U) 4 0. But it is easy to check that Ann(U) is a d-ideal of R. Hence U and Ann(U) are non-zero d-ideals of R such that U Ann(U) = 0. This contradicts the fact that R is d-prime.

Therefore J is a non-zero ideal of R contained in I.

Now we show (by induction on k) that dk(J) U k ~= I for all k > 0. For k = 0 this has already been established. Let k > 0 and assume that d k - l ( J ) U k - l ~ I. Then; for j s J, u l , . . . , u k ~ U; we have

i.e.

[dk- l(j) Ul " " " Uk_ I~ Uk] El,

d~- *(j) u, ... u~_ l d(u~) - d~(j) ui " " u~

k 1

- ~ dk-~(j) u ~ . . . u ~ _ ~ d ( u 3 u ~ + ~ . . . u k e I .

i = 1

Vol. 45, 1985 Lie structure of a ring with a derivation 333 Since U is a d-ideal in R and d k- i (j) U k- i c= I, it follows that

d k(j) U l'" Uk ~ I.

Hence dk(j) U k ~ I for all k => 0.

Since R is noetherian there exists s >= 0 such that

J'= J + d(J) + ... + d~(J) = J + d(J) + ... + d~+~(J) is a d-ideal of R (see Lemma 1.1).

Let J " = J ' U s. Then J " is a non-zero d-ideal of R contained in I. This completes the proof.

As an immediate consequence of Theorem 3.4 we get

Corollary 3.5. Let 6 ~ 3 (R), R be noetherian d-prime, and A be a non-zero Lie ideal of R o. Then every non-zero Lie ideal of the Lie ring A contains a non-zero d-ideal of R.

4.

The sequence

{R,}. Let R o ~ R1 ~ R 2 ~ . - - be the normal sequence of Lie subrings of R o defined by R,+ 1 = [R,, Ro] for n = 0, 1, . . . , and assume that R is 2-torsion-free.

C. R. Jordan and D. A. Jordan showed that if R is domain or R is noetherian d-prime then Ra is a prime Lie ring ([4] Theorems 4 and 5), and if R is noetherian d-prime then every non-zero Lie ideal of Ra contains a aaon-zero d-ideal of R ([4] Theorem 6).

The aim of this section is to prove analogues of the above facts for each Lie ring R,.

Observe that, since each R , is a Lie ideal of R o, we obtain such analogues at once from Corollaries 2.7 and 3.5, but with the assumption that 6 ~ 3(R). However we show, using an idea from [4], that it suffices to assume that 2 ~ 3(R).

Proposition

4.1. Let 2 ~ 3(R). If R is quasi-prime then each R, is a non-zero prime Lie ring.

P r o o f. Let n be a fixed natural number. It is clear, by Corollary 2.6, that R , 4: 0. Let I and J be Lie ideals of R , such that [I, J] -- 0. Let i ~ I, j ~ J, r ~ R.

Since 2" RR, ~ R,, it follows that [i, 2"jr] ~ I, so that 2"ijd(r) ~ I. Therefore - 2"j2(d(i)d(r) + id2(r)) = [2"ijd(r),j] = 0,

that is, j2(d(i) d(r) + idZ(r)) = O.

Using Proposition 1.3 and Lemma 2.2, proceed as in the proof of Theorem 5 in [4] to obtain I = 0 or J = 0.

Proposition

4.2. Let R be noetherian d-prime, 2 ~ 3(R), and let n be a natural number.

Then every non-zero Lie ideal of R, contains a non-zero d-ideal of R.

P r o o f . Let I be a non-zero Lie ideal of R n. Then, by Propositions 1.5 and 4.1, [I, I] 4: O. Choose a, b ~ I such that [a, b] 4: 0. Let c = 2" + i [a, b], J = c R, and L = 2" R n.

Observe that J ~ I. In fact, if r ~ R then

cr = 2-2"r[a, b] = [a, 2"rb] + [2"ra, b] ~ I, since 2" R R, ~= R,.

It is easy to show that L 2 ___ L and d(L) ~= L.

Further we proceed as in the proof of [4] Theorem 6.

A d-ideal P of R is said to be quasi-prime if the ring RIP is quasi-prime. A Lie ideal of a Lie ring L is called prime if LIP is a prime Lie ring.

Using the above propositions and arguments from the proofs of Corollaries 1 and 2 in [4] we can prove the following

Proposition

4.3. Let 1/2 ~ R and n be a natural number. If P is a quasi-prime ideal of R such that d(R) 4: P then P c~ R, is a prime Lie ideal of R n. Furthermore, if R is noetherian, every prime Lie ideal of R n is of the form P c~ Rn for some quasi-prime ideal P of R such that d(g) 4: P.

We end this paper with the following observations.

Assume that 1/2 ~ R. Then it is easy to check that each R, is a d-ideal of R. Moreover, we have

R~ = Rd(R),

R 2 = Rd2(R) + Rd(R) 2 = Rd2(R),

R 3 = Rd3(R) + Rd2(R) d(R) + Rd(R) 3 = Rd3(R) + Rd2(R) d(R), and more generally, by induction, we obtain the following

Proposition

4.4. If 1/2 ~ R then

R, = Z ~ gdil(R) d~2(R) "'" di~(R),

k<=n i~+ " + i k = n

for any n > O.

In particular, (Rd(R))" ~ R, and Rd"(R) ~ R,.

As an immediate consequence of the above proposition we get

Corollary

4.5. If 1/2 ~ R then for any natural n and m R,R,. ~ R,+m.

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Anschrift des Autors:

Andrzej Nowicki Institute of Mathematics N. Copernicus University uL. Chopina 12/18 87-100 Torufl Poland

Eingegangen am 1.6. 1984