Math. Z. 136,245-248 (1974) 9 by Springer-Verlag1974 A Note On Derivations of Commutative Rings Surender K. Gupta Introduction Let B/A be an extension of commutative rings of prime characteristic p > 0. In w1, we show that if B/A is a Galois extension then any restricted B-submodule of DerB/a which is a B-direct summand is a Lie subring. This generalises Theorem 1 of Ojanguren and Sridharan [33 (which asserts that any restricted subspace of the derivation algebra of a fie!d of prime characteristic is a Lie subring), it also shows that the Lie subring condition in the Galois theory of Yuan [4] is not necessary. Moreover, we give an example to show that a restricted submodule in general need not be a Lie subring. In w 2, we show that if B has a p-basis over A, then there is a bijection between the set of subextensions C/A of B/A such that B has a p-basis over C and restricted submodutes of DerB/a which are B-direct summands and satisfy an additional condition. We give an example to show that this additional assumption cannot in general be removed. The author is indebted to Professor R. Sridharan and Dr. Amit Roy for encouragement and helpful suggestions and to the referee for many helpful suggestions which led to an improvement of the earlier version of the paper and also to simplification of the proof of Example 2.3. w1. Restricted Submodules Let B/A be a commutative ring extension of prime characteristic p > 0. We call it a Galois extension (as in [4, p. 165]) if i) B is a finitely generated projective A-module, ii) B p c A, iii) Given any P E spec(B),B e admits a p-basis over A~ where p = P c~ A. Theorem 1.1. Let B/A be a Galois extension. Let D be a restricted B-submodule of DerB/a which is a direct summand of DerB/a as a B-module. Then D is a Lie subring. Proof Since a submodule of a restricted Lie algebra is a subalgebra if all its localisations are, we may and do assume that B is local. Further, since D is a direct summand of DerB/a, it follows that D is a free B-module,