proceedings of the american mathematical society Volume 90. Number I, lanuary 1984 ON THE DERIVATION OF x " IN A RING DOUGLAS BRIDGES AND JEFFREY BERGEN Abstract. The main result of this paper gives conditions which ensure that an additive map of a ring R into itself which satisfies a generalisation of the rule áx "'/ax = nx"~ is a derivation on R. It is well known that the product rule for derivatives can be proved using only the additivity of the differentiation process, the Chain Rule, and the formula for the derivative of the map x -» x2 (see Lemma 1). In this note, we show how this generalises to a characterisation of derivations on certain rings. Let R be a ring, and recall that a derivation on R is an additive mapping D: R -» R such that D(xy) = xD(y) + D(x)y (x, y G R). It is easy to show that for each integer n > 2, such a map has the nth power property: n D(x") = 2 x"-JD(x)xJ-x (x G R). 7=1 (We adopt the convention that x°y — y — yx° for all x, y in R.) Conversely, if D: R -» R is an additive map with the n th power property for some n > 2, under what conditions is D a derivation on R1 One almost trivial case when D is a derivation is dealt with in our first result. Lemma 1. If R is a commutative ring with no 2-torsion, and the additive map D: R — R has the second power property, then D is a derivation on R. Proof. For all x, y in R, 2(x+y)(D(x) + D(y)) = d{(x + yf) = D(x2) + 2D(xy) + D(y2) = 2xD(x) + 2D(xy) + 2yD(y). It readily follows that D(xy) = xD(y) + D(x)y. D A more general answer to the question above was given by Herstein [1]. For convenience, we state as much of Herstein's result as we shall need. Theorem 1. Let R be a prime ring that is not of characteristic 2, and let D: R -* R be an additive map with the 2nd power property. Then D is a derivation on R. For our main result we need two more lemmas. Received by the editors November 10, 1982 and, in revised form, January 5, 1983 and April 22, 1983. 1980 Mathematics Subject Classification. Primary 16A72. ©1984 American Mathematical Society 0002-9939/84 $1.00 + $.25 per page 25 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 26 DOUGLAS BRIDGES AND JEFFREY BERGEN Lemma 2. Let R be a ring, D an additive map of R into itself, and y an element of R such that both y and D(y) are in the center of R. For each x in R and each positive integer k, define fk(x)=2xk-W(x)x^-x. 7 = 1 Then (i) f„(x + y)= 2 (")yn-JfJ(x) + "(x + y)n~]D(y) (n>\). >=1 W ' Proof. We proceed by induction on n. In the case n = 1, equation (1) is just a particular case of the additivity of D. Now suppose that we have proved (1) for n = k. Then fk+\(*+y) = (x+y)fk(x+y) + D(x+y)(x+y)k = (x+y) 2 (kl)yk-Jfj(x) + k(x+y)k-]D(y) 7=1 \J ' + D(x)(x+y)k + D(y)(x+y)k + k(x+y)kD(y) + D(x)(x+y)k +(x+y)kD(y) = 2 (kj)[fJ+Á*)-D(*)xJ]yk-J + i(k)yk+i-yfj(x) A- + (k+l)(x+y)kD(y)+ D(x)2 (fa rJyk-j 7=0 =/*+,(*)+ 2 k+ x-Jf(x) + kykfx(x) 7 = 2 AM, + (k+l)(x+y)kD(y) + D(x)yk =fk+l(x)+22(ky)yk+i-Jfj(x) + (k + l)ykfx(x) +(k + l)(x+y)kD(y) = *2 (k +i l )yk+i-Jfj(x) +(k+ 0(* + ^)^(^). D 7=1 V J I In the proofs below, for a fixed integer «>2we need the observation that the (n — l)-by-(« — 1) matrix A with (k, j)lh entry kJ(") has determinant 8=à0tA= l"ff(£))(»-o* n ('-v). *=i l<i<7'<n—1 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE DERIVATION OF x" IN A RING 27 Lemma 3. Let R be a ring, n > 2 an integer, D: R -» R an additive map with the nth power property, and y an element of R such that both y and D(y) are central. Suppose that for each x in R,8x — 0 entails x — 0. Then k (2) D{xky"~k) =y"~k 2 xk~JD(x)xJ~x + (n - k)xkyn~k~xD(y) 7=1 whenever x G R and k = 1,2,... ,n — 1. Proof. For each jc in /? and each positive integer &, define /*(*) = 2 xk~JD(x)xJ-x. 7=1 By the «th power property and Lemma 2, D((x+y)n)=fn(x+y) = fn(x)+2(ni)yn-JfJ(x) + n(x+yy-lD(y). 7=1 W ' On the other hand, D((x+y)")=D^^(nj)y^xA = D(x")+ 2 (nj)D(y"-JxJ) + D(y"). 7=1 V ' Equating these expressions for D((x + y)"), we obtain (3) 2 (nj)[D(y"-JxJ) -y"-J/j(x)] + D(y") - n(x + y)"''D(y) = 0. Since y commutes with D(y), we have D(y") — ny"~xD(y), and therefore, n(x+y)"-XD(y)-D(y")^n^ ( " Mx»"'1"^) 7=1 V 7 ' 2 («-»(" j^"-1-7^)- 7=1 From this and (3) we have »i-i 2 Cj)[D(xJy"-J) -yn-jfj{x) - (n - j)x^y"-x^D(y)] = 0. Replacing x by kx, we now obtain 2 kJ(Hj)[D(x'y*-J) -y"-Jfj(x) - (n -j)xY~l-JD(y)] = 0 (k= 1,2,...,«- 1). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 28 DOUGLAS BRIDGES AND JEFFREY BERGEN Since the matrix of these equations has (k, j)th entry kJ(k), for k = 1,2,... ,n — I, we have s[D{xJy"~J) -y"-Jfj{x) - (n - j)x^y"-x^D(y)] = 0 from which the desired conclusion follows. D Theorem 2. Let R be a ring, n > 2 an integer, and D: R -» R an additive mapping with the nth power property. Then D is a derivation under each of the following circumstances: (i) R is commutative with an identity e, and for each x in R,8x = 0 entails x = 0, (ii) R is a prime ring with an identity e and with characteristic c, where c — 0 or c > n, (iii) R is a commutative integral domain with characteristic c, where c = 0 or c > n. Proof. First note that in all these circumstances we have x = 0 whenever x G R and 8x = 0. In cases (ii) and (iii), this is clear if c = 0; while if c is prime, it follows from the fact that as c > n, c does not divide 8. Consider cases (i) and (ii). Since D(e) = D(e") = ne"~xD(e) — nD(e), we have (n — l)D(e) = 0, and therefore 8D(e) = 0. Hence D(e) — 0. If we now apply Lemma 3, taking y = e and k = 2 in (2), we see that D satisfies the 2nd power property. It follows from Lemma 1 in case (i), and from Theorem 1 in case (ii), that D is a derivation. Now consider case (iii), and apply Lemma 3. Taking k = n — 1 in (2), we see that for arbitrary x and y in R, »7-1 (4) D(x"-Xy) =y 2 xn~x-JD(x)xJ-x + xn~xD(y) 7=1 = (n - l)x"-2yD(x) + x"~xD(y). With>> = x"+x, we obtain nx2"-2D(x2) = D((x2)") = D(x"-Xx"+X) = (n- \)x2"-xD(x) + x"-xD(x"+x). Also, putting y = x2 in (4) gives D(x"+X) = (n- l)x"D(x) + x"-xD(x2). Hence nx2"-2D(x2) = (n - l)x2n~xD(x) + x"~l[(n - l)xnD(x) + x"-xD(x2)] and so (n - l)x2"-2[D(x2) - 2xD(x)] = 0. Since R has no zero divisors, it follows that D(x2) — 2xD(x). (Note that Z)(0) = 0, as D is additive.) As x is arbitrary, D has the 2nd power property, and is therefore a derivation, by Lemma 1. D License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE DERIVATION OF x" IN A RING 29 Acknowledgment. The authors wish to thank the referee for his suggestions, which greatly improved the presentation of the paper. References 1. I. N. Herstein, Joráan áerivations of prime rings, Proc. Amer. Math. Soc. 8 (1957), 1104-1110. Department of Mathematics, University College, Buckingham MK.18 1EG, England Department of Mathematics, DePaul University, Chicago, Illinois 60604 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use.