Differential identities in the theory of inverse problems for kinetic equations

For an arbitrary associative commutative ring $L$, we establish an identity of a certain type that relates arbitrary finite families of elements of the ring and its differential operators. When $L$ is an algebra of functions defined on a manifold $M$, and the differential operators are vector fields, one can derive from the identity established some known identities that can be used to prove uniqueness theorems in the theory of inverse problems for kinetic equations. In some cases, we are able to give necessary and sufficient conditions for the existence of the identity.