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A well-known theorem factors a scalar coefficient differential operator givena linearly independent set of functions in its kernel. The goal of this paperis to generalize this useful result to other types of operators. In place ofthe derivation $\partial$ acting on some ring of functions, this paperconsiders the more general situation of an endomorphism $\mathfrak{D}$ actingon a unital associative algebra. The operators considered, analogous todifferential operators, are those which can be written as a finite sum ofpowers of $\mathfrak{D}$ followed by left multiplication by elements of thealgebra. Assume that the set of such operators is closed under multiplicationand that a Wronski-like matrix produced from some finite list of elements ofthe algebra is invertible (analogous to the linear independence condition).Then, it is shown that the set of operators whose kernels contain all of thoseelements is the left ideal generated by an explicitly given operator. In otherwords, an operator has those elements in its kernel if and only if it has thatgenerator as a right factor. Three examples demonstrate the application of thisresult in different contexts, including one in which $\mathfrak{D}$ is anautomorphism of finite order.

On Factoring an Operator Using Elements of its Kernel