Solutions for PDEs with constant coefficients and derivability of functions ranged in commutative algebras

It is well known that the real and imaginary parts of any holomorphic function are harmonic functions of two variables. In this paper,
we extend this idea to finite-dimensional commutative algebras; that is, we prove that if some basis of a subspace of a commutative
algebra satisfies a polynomial equation, then the components of a monogenic function on the subspace are solutions of the respective
partial differential equation (PDE). We illustrate these concepts with a few examples.