&lt;p&gt;Let n be a positive integer, R a prime ring, U a nonzero right ideal, and d a derivation on R. Under appropriate additional&#12288;hypotheses, we prove that if d&lt;sup&gt;n&lt;/sup&gt;(U) is finite, then either R is finite or d is nilpotent. We also provide an extension to semiprime rings.&lt;/p&gt;</p

Bell, Howard E.

Okayama University Scientific Achievement Repository

Mathematical Journal of OkayamaUniversityVolume 41, Issue 1 1999 Article 2JANUARY 1999Higher Derivatives and Finiteness in RingsHoward E. Bell∗∗Brock UniversityCopyright c©1999 by the authors. Mathematical Journal of Okayama University is produced byThe Berkeley Electronic Press (bepress). http://escholarship.lib.okayama-u.ac.jp/mjouHigher Derivatives and Finiteness in RingsHoward E. BellAbstractLet n be a positive integer, R a prime ring, U a nonzero right ideal, and d a derivation on R.Under appropriate additional&#x3000;hypotheses, we prove that if dn(U) is finite, then either Ris finite or d is nilpotent. We also provide an extension to semiprime rings.Math. J. Okayama Univ. 41 (1999), 21{25HIGHER DERIVATIVES AND FINITENESS IN RINGSHOWARD E. BELLAbstract. Let n be a positive integer, R a prime ring, U a nonzeroright ideal, and d a derivation on R. Under appropriate additionalhypotheses, we prove that if dn(U) is ¯nite, then either R is ¯nite ord is nilpotent. We also provide an extension to semiprime rings.In [2] it is proved that if R is a prime ring and d is a derivation on Rsuch that d(R) is ¯nite, then either R is ¯nite or d = 0. This result invitesan investigation of prime rings with derivation such that dn(U) is ¯nite forsome derivation d; some n ¸ 1, and some ideal (or right ideal) U . If U is anonzero ideal, or if U is a nonzero right ideal and R is suitably-restricted,we can show that either R is ¯nite or d is nilpotent on R.1. PreliminariesLet R be a ring and S a nonempty subset of R, and let f be a mappingfrom R to R. We say that f is nilpotent on S if fn(S) = f0g for somepositive integer n; more generally, we call f periodic on S if there existdistinct positive integers m, n such that fn(x) = fm(x) for all x 2 S. Wedenote the right annihilator of S by Ar(S).We begin by stating and proving a lemma from [1].Lemma 1.1. An in¯nite prime ring contains no nonzero ¯nite rightideal.Proof. Let R be in¯nite and prime, and suppose H is a nonzero ¯niteright ideal. Let H n f0g = fx1; x2; :::; xng. For each i = 1; 2; :::; n, de¯nefi : R! H by fi(r) = xir for all r 2 R. Then fi(R) is ¯nite, hence ker fi =Ar(xi) is a right ideal of R having ¯nite index in R. Thus K =Tni=1 ker fiis a right ideal of ¯nite index, necessarily nonzero, such that HK = f0g.But this cannot happen in a prime ring.It is well-known that if R is a ring of prime characteristic p and d isa derivation on R, then dp is also a derivation. This observation is the keyto the following lemma, which we shall use several times.Supported by the Natural Sciences and Engineering Research Council of Canada,Grant No. 3961.21 1Bell: Higher Derivatives and Finiteness in RingsProduced by The Berkeley Electronic Press, 199922 HOWARD E. BELLLemma 1.2. Let n be a ¯xed positive integer, and let R be a classof prime rings with the following property:(*) If R 2 R admits a nonzero derivation d such that d(U) is ¯nitefor some nonzero ideal (resp. right ideal) U , then R is ¯nite.Then for any R 2 R and any derivation d such that dn(U) is ¯nite forsome nonzero ideal (resp. right ideal) U , either R is ¯nite or d is nilpotenton R.Proof. It will su±ce to prove the right ideal version. Let R 2 R and Ua nonzero right ideal of R, and let d be a derivation on R such that dn(U) is¯nite. If charR = 0, then dn(U) = f0g; and by a result of Chung and Luh[4], d is nilpotent on R. Thus, we assume that R has prime characteristicp. Let P be the smallest power of p which is at least n, and let ± = dP :Since ± is a derivation and ±(U) is ¯nite, it follows from (¤) that either Ris ¯nite or ± = 0; and the latter possibility implies that d is nilpotent onR.2. The case of U an idealIf U is assumed to be an ideal, then we can show that dn(U) can be¯nite only in the obvious ways.Theorem 2.1. Let n be a ¯xed positive integer. Let R be a primering and d a derivation on R such that dn(U) is ¯nite for some nonzeroideal U . Then either R is ¯nite or d is nilpotent on R.Proof. Let R be any prime ring, U any nonzero ideal and d a deriva-tion on R such that d(U) is ¯nite. Consider the map © : U ! d(U) givenby ©(x) = d(x) for all x 2 U . Then ker© = fx 2 U j d(x) = 0g is asubring of U of ¯nite index in U , so by a result of Lewin[5], ker© containsan ideal H of U which has ¯nite index in U . If H = f0g, then U is ¯nite;and by Lemma 1.1, R is ¯nite. Suppose, then, that H 6= f0g. For all x 2 Uand y 2 H, we have 0 = d(yx) = yd(x) + d(y)x = yd(x); and thereforeyUd(U) = f0g. But for y 2 H nf0g; yU is a nonzero right ideal of R, henceAr(yU) = f0g. Thus d(U) = f0g, and it follows easily that d = 0. Ourresult now follows by Lemma 1.2.3. The case of U a right idealMost of the proof of Theorem 2.1 works if U is assumed to be onlya right ideal; the hypothesis that U is a two-sided ideal is used only inshowing that y 2 H n f0g implies yU 6= f0g. Of course, if R is a domain,the same implication holds; hence, we have2Mathematical Journal of Okayama University, Vol. 41 [1999], Iss. 1, Art. 2http://escholarship.lib.okayama-u.ac.jp/mjou/vol41/iss1/2HIGHER DERIVATIVES AND FINITENESS IN RINGS 23Theorem 3.1. Let R be a ring with no nonzero divisors of zero, andU a nonzero right ideal of R. If d is a derivation on R and dn(U) is ¯nitefor some positive integer n, then either R is ¯nite or d is nilpotent on R.By combining Theorem 2.1 and a result in [3], we obtainTheorem 3.2. Let R be a prime ring and U a nonzero right idealof R. If d is a nonzero derivation and there exists a positive integer n forwhich dn(U) is ¯nite and central, then d is nilpotent on R.Proof. Assume d is not nilpotent. Then by the ¯nal result in [3], R iscommutative and hence U is an ideal. By Theorem 2.1, R is ¯nite, hencea ¯nite commutative domain - i.e. a ¯nite ¯eld. But it is known that ¯nite¯elds admit no nonzero derivations.Whether we can always replace U in Theorem 2.1 by a right ideal isan open question; however, we do have an a±rmative answer for PI-rings.Theorem 3.3. Let R be a prime PI-ring, and let d be a derivationon R such that dn(U) is ¯nite for some nonzero right ideal U and somepositive integer n. Then either R is ¯nite or d is nilpotent on R.Proof. In view of Lemma 1.2 and its proof, we may assume that d(U)is ¯nite and R has prime characteristic p. It is well known that a prime PI-ring has nonzero center Z; and if z 2 Z n f0g, then d(zp) = pzp¡1d(z) = 0,so R has nonzero central constants.Suppose that d(U) 6= f0g, and let jd(U)j = k. Then for any non-constant u 2 U and nonzero central constant z, there exist distinct m,n 2 f1; 2; :::; k + 1g such that d(zmu) = d(znu) - i.e. (zm ¡ zn)d(u) = 0;and since Z has no elements which are zero divisors in R, we get zm = zn.It follows easily that there exist distinct integers M , N such that zM = zNfor all central constants z, hence Z satis¯es the identity zMp = zNp andtherefore Z is a ¯nite ¯eld.Since R is a prime PI-ring, its central localization RZ is a primitivePI-ring [6, Theorem 6.1.30]. Moreover, since Z is a ¯eld, R »= RZ andhence R is primitive. By a classical result of Kaplansky, R is therefore¯nite-dimensional over Z; hence R is ¯nite.In the proof of this theorem, the right ideal property of U is used onlytwice: in the proof of Lemma 1.2, to show that d nilpotent on U impliesd nilpotent on R, and in the argument above to guarantee that ZU µ U .Thus, our methods yieldTheorem 3.4. Let R be a prime PI-ring and S an additive subgroupsuch that ZS µ S. If R admits a derivation d such that dn(S) is ¯nite,then either R is ¯nite or d is nilpotent on S.3Bell: Higher Derivatives and Finiteness in RingsProduced by The Berkeley Electronic Press, 199924 HOWARD E. BELL4. A theorem on semiprime ringsWe conclude the paper with a theorem which replaces \nilpotent" by\periodic", and which is available in the setting of semiprime rings.Theorem 4.1. Let R be a semiprime ring having no nonzero ¯niteright ideals. If U is a nonzero right ideal of R and d is a derivation onR such that dn(U) is ¯nite for some positive integer n, then U contains anonzero right ideal U1 of R such that d is periodic on U1.The proof uses a rather general lemma.Lemma 4.2. Let R be an arbitrary ring and S a nonempty subset ofR. If f : R ! R is a mapping such that f(S) µ S and fn(S) is ¯nite forsome positive integer n, then f is periodic on S.Proof. Since f(S) µ S, for each positive integer k we have fk+1(S) =fk(f(S)) µ fk(S). Thus, if fn(S) is ¯nite, the chain fn(S) ¶ fn+1(S) ¶fn+2(S) ¶ : : :must become stationary at some point, say at fN (S) =fx1; x2; : : : ; xmg. Then for each u ¸ 1, the ordered m-tuple(fu(x1); fu(x2); :::; fu(xm)) is a permutation of (x1; x2; :::; xm). Thereforethere exist distinct u, v ¸ 1 such that fu(xi) = fv(xi) for all i = 1; 2; :::;m.Now for each x 2 S, fN (x) = xi for some i = 1; 2; :::;m; thereforefN+u(x) = fN+v(x) for all x 2 S.Proof of Theorem 4.1. Let U be a nonzero right ideal with dn(U)¯nite. Let T be the torsion ideal of R; and for each prime p, let Tp bethe p-primary component of T . If T = f0g, then dn(U) = 0, so clearlyd is periodic on U . If T 6= f0g and U \ T = f0g, then UT = f0g; andit follows easily by semiprimeness that TU = 0 as well. It follows thatUdm(U) = f0g = dm(U)U for all m ¸ n. By applying d to these equationsrepeatedly, we see that di(U)dj(U) = f0g for all nonnegative i, j withi ¸ n or j ¸ n. By Leibniz' formula, we obtain d2n¡1(U2) = f0g, hence dis periodic on U2.The remaining case is that of U \T 6= f0g, in which case U \Tp 6= f0gfor some prime p. Now by semiprimeness of R, pTp = f0g; thus, V = U\Tpis a nonzero right ideal of R with pV = f0g. Moreover, dP (V ) is ¯nite,where P is the smallest power of p which is at least n.It remains only to prove that if V is any nonzero right ideal with pV =f0g and dp® (V ) ¯nite for some ®, then d is periodic on some nonzero rightideal contained in V . We use induction on¯¯¯dp®(V )¯¯¯. A crucial observationis that, by Leibniz' formula,4Mathematical Journal of Okayama University, Vol. 41 [1999], Iss. 1, Art. 2http://escholarship.lib.okayama-u.ac.jp/mjou/vol41/iss1/2HIGHER DERIVATIVES AND FINITENESS IN RINGS 25dp®(xy) = dp®(x)y + xdp®(y) for x; y 2 R with at least one of x; y in V:(1)If¯¯¯dp®(V )¯¯¯= 1, then dp®(V ) = f0g = dp®+1 (V ), so d is obviouslyperiodic on V . Now assume the result holds for nonzero right ideals bVwith pbV = f0g and ¯¯¯dp® (bV )¯¯¯ < k, and let V be a nonzero right idealwith pV = f0g and¯¯¯dp®(V )¯¯¯= k. If V contains a nonzero right idealI of R with¯¯¯dp®(I)¯¯¯< k, the desired conclusion is immediate from theinductive hypothesis; hence we assume that for every nonzero right idealI contained in V , dp®(I) = dp®(V ). Now since V is in¯nite and dp®(V ) is¯nite, V contains a nonzero subset S such that dp®(S) = f0g; and since Ris semiprime, for s 2 S n f0g, sR is a nonzero right ideal contained in V .Therefore, by (1) we get dp®(V ) = dp®(sR) = sdp®(R) µ V ; hence dp® isperiodic on V by Lemma 4.2. Thus, d is periodic on V .References[1] H. E. Bell and A. A. Klein, On ¯niteness, commutativity, and periodicity in rings,Math J. Okayama Univ. 35 (1993), 181-188.[2] H. E. Bell and A. A. Klein, Ideals contained in subrings, Houston J. Math. 24 (1998),1-8.[3] H. E. Bell, A. A. Klein, and J. Lucier, Nilpotent derivations and commutativity,Math. J. Okayama Univ. 40 (1998), 1-6.[4] L. O. Chung and J. Luh, Nilpotency of derivatives on an ideal, Proc. Amer. Math.Soc. 90 (1984), 211-214.[5] J. Lewin, Subrings of ¯nite index in ¯nitely-generated rings, J. Algebra 5 (1967),84-88.[6] L. H. Rowen, Ring Theory I and II, Academic Press 1988.Howard E. BellDepartment of MathematicsBrock UniversitySt. Catharines, OntarioCanada, L2S 3A1(Received November 26, 1999)5Bell: Higher Derivatives and Finiteness in RingsProduced by The Berkeley Electronic Press, 1999

Higher Derivatives and Finiteness in Rings

Abstract

Similar works

Full text

Available Versions

Okayama University Scientific Achievement Repository

Okayama University Digital Information Repository