Loy, Richard J.

English

Loy Richard J.

University of Szeged

A note on the existence of derivations By R. J. LOY*) in Clayton (Victoria, Australia) Let K be a commutative ring with unit; 91, © two i£-algebras, and denote by, £(91,93) the (left) ^-module of all ^-linear mappings of 91 into 93. We write £(91) for £(5t, 91) and note that this is a ^-algebra. If <p, D 6 £(91, S ) and D satisfies the equation D(xy) = Dx-(py + (px-Dy for all x, y6 91, we call D a (^-derivation. If 91 is a subalgebra of 93 and If £(91, 93) is the identity map on 91, an /-derivation will be termed a derivation. We will only consider cp-derivations where cp is a homomorphism. Note that our ^-derivations are different from the (^-derivations of AMITSUR [L]I and JACOBSON [3], which are additive mappings defined on fields satisfying D(xy)= = q>x-Dy + Dx-y .(or D(xy) = Dx-cpy + x-Dy) for all x, y in the field. Our ^-deriv-ations satisfy the defining equation for the ( f , <p)- derivations of [4], p. 177; they are closely related to the (^-derivations of order one of [5]. Now let D £ £(21,93) be a (^-derivation of 91 into © for some homomorphism <pfQ (21, 93). We define the 'spread' <» of the D as the smallest subalgebra of 93. contain-ing both the range of D and the range of cp. S is in fact the smallest subalgebra of 93 such that both cp and D are in £(91, <3), that is, such that D is a (^-derivation from 91 into S . It is clear that <2 depends on the mapping <p, and if D is a (¡»-deriv-ation for more than one <p it may have more than one spread. In the cases where we use this notion of spread however, it will be clear which mapping cp is being considered, and no explicit mention of it will be made. We will denote the algebra direct sum of two A'-algebras 91, © by 91©©, and their AT-module direct sum by 91+©. Thus 91©© is the X-algebra of all pairs (a, b) with 91, ¿ 6 © and componentwise operations, while 9 I + S is the AT-module of all such pairs and will not be assumed to have the algebra structure of 9Iffi©> *) The author is a General Motors—Holden's Research Fellow. 90 R. J. Loy If (¡¡>£.2(31, ®) is a homomorphism, the A"-linear isomorphism of 21 into IHffi® defined by ' <Px = (x, (px) for all x € 9 i , will be called the (/»-embedding of 91 in 9iffi®. Any isomorphism of this type will be termed an embedding of 21 in 91©®. With these notations we have the following result. T h e o r e m 1. Let 21 be a K-algebra with a (K-module) direct sum decomposition 9 [ = ® +3 where 3 is a proper left ideal of 21 and ® is a subalgebra of 91 but not a left ideal. Then there is an embedding <P of 21 in 91 © £(91), such that 91 admits a nonzero ^-derivation D : 21 —2t©£(2t) satisfying Z>(®) = {0}. P r o o f . For 21, let a = x + y be the unique decomposition of a with x € ® , and define idempotent.mappings P, g £ £ ( 9 I ) by Pa = x, Qa=y. Then we have immediately that P2 = P, PQ = QP = 0, Q2 = Q and P + Q = I. For a £ 21 denote by a the image of a in £(21) under the left regular representation and define A"-linear mappings <p, A from 21 into fi(2I) by (pa = PaP + QaQ and Aa = QaP for all a£ 21. Since 3 is a left ideal it is invariant under each a for a£ 21 and so PaQ = 0 for any a £ 21. But then if x, j> 6 91, (x v) = x y and so cp(xy) = Px(P+Q)yP + Qx(P + Q)yQ = PxPyP + QxQyQ = .= (PxP + QxQ) (PyP.+ QyQ) = (px-cpy. Also A(xy) = Qx(P + Q)yP = QxPyP + QxQyP = = QxP{P + Q)y(P + Q) + {P + Q)x(P + Q)QyP = Ax-<py + <px-Ay. Thus <p is a homomorphism and A is a ^-derivation, which furthermore is non-zero since ® is not a left ideal. We now make use of a construction of SINGER and WERMER [6]. Let <P be the «^-embedding of 91 in 2l©fi(2l) and define a ^-linear mapping D: 2i—2tffi£(2t) by Da = (0, Aa) for all 2f. It is easily seen that D is a «^-derivation, non-zero since A is non-zero. If x € ® then (xP)y£23 for any j € 2 I , so that (QxP)y = 0. Thus J x = 0 if x € ® , that is, i)(®) = {0}. The reason for wanting to be an isomorphism is that we can identify 2i with $(21) to get the following result. C o r o l l a r y 1. Let 21 satisfy the conditions of the theorem. Then 91 admits non-zero derivation D into an extension of 91 such that D0B) = {0}. The spread of (the (¡»-derivation) D is in general non-commutative even when 21 is commutative1) . A necessary and sufficient condition for the spread to be commutative is given by the following result. ') The range of D is easily seen to be a zero ring. Existence of derivations 91 T h e o r e m 2. If 91 is commutative then the spread of D is commutative if and only if 3 2 © =0 . P r o o f . From the definition of D, the spread of D is commutative if and only if the spread of A is commutative. Now the spread of A is the algebra generated by the set {(pa, Aa: a€91}. Since 91 is commutative and cp is a homomorphism <px-<py = cpy-<px for all x, Also, f rom the definitions, Ax-Ay = Ay-Ax = 0. Thus it suffices to consider necessary and sufficient conditions for Ax-(py = (py-Ax, that is, QxPyP = QyQxP for all x, j £ 91. (i) Necessity of stated condition. Let x, y£~s, z £ © . Then (QxPyP)z — =(QxPy)z = 0 since y(z)=yz£~s and P(3) = {0}. On the other hand, (QyQxP)z= = (QyQx)z = (QyQ)xz = (Qy)xz = Q(yxz)=yxz. Thus the condition is necessary. (ii) Sufficiency of stated condition. If z £ 3 then (QxPyP)z = 0 = (QyQxP)z for any x, >>€9t since Pz = 0. Since 91 = © + 3 it thus suffices to consider z € © . Thus suppose z £ © , and let x,y£91 have decompositions x = u1+vi, y = u2 + v2; f ° r /"=1,2. Then the decompositions of xz,yz,xu2z are u^ + v^z, u2z + v2z, ulu2z + vlu2z, respectively. Thus (QxPyP)z — (QxP)(u2z+ v2z) = Q(xu2z) = vlu2z, and (.QyQxP)z = {QyQ){uiz + viz) = Q(>'i'1z) = yv^z. The difference between these is yviz—'viu2z = v1z(y — u2) = v1v2z = 0, since 'v1,v2£3> zg©. Thus the condition is sufficient. - It follows that the extension of 91 in Corollary 1 may be taken to be commutative if 91 is commutative and 3 2 © = 0 . C o r o l l a r y 2. Let 9T be a commutative K-algebra with 9 I 2 = 9 i . 2) Suppose that 91 has a direct sum decomposition 9 t = © + 91 where S is a subalgebra and 91 is a nontrivial nilpotent ideal. Then 91 admits a non-zero derivation into an extension algebra of 91 which annihilates If 9? is a zero ring or if the sum is an algebraic direct sum then the extension algebra may be taken to be commutative. P r o o f . By hypothesis © is a subalgebra, so using Corollary 1 it-suffices to show that it is not an ideal. Supposing to the contrary, we have S S R ^ S and so 9I2$i© + 9i2 whence SR2 = 9L But this is impossible since 5R is non-trivial and nil-potent. The last statement is clear from Theorem 2. Acknowledgements The idea of Theorem 1 stemmed from a perusal of the matrix proof of the theorem of MASCHKE in the theory of group representations. See, for example, Theorem 16. 3. 1 of [2]. The author would like to thank Professor J. B. MILLER for helpful discussions concerned with this paper. 2) This is true, for instance, if 91 has an identity. 92 R. J. Loy: Existence of derivations References [1] S . A . AMITSUR, A generalization of a theorem on linear differential equations, Bull. Amer. Math. Soc., 54 (1948), 937—941. [2] M . H A L L , The theory of groups (New York, 1 9 6 5 ) . [3] N. JACOBSON, Pseudolinear transformations, Ann. of Math., 38 (1937), 484—507. [4] Lie algebras (New York, 1962). [5] H . OSBORN, Modules of differentials. I , Math. Annalen, 1 7 0 ( 1 9 6 7 ) , 2 2 1 — 2 4 4 . [6] I. SINGER and J. WERMER, Derivations on commutative normed algebras, Math. Annalen, 129 (1955), 260—264. (Received February 20, 1968) MONASH UNIVERSITY CLAYTON, VICTORIA AUSTRALIA 

A note on the existence of derivations

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A note on the existence of derivations

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