The idea goes back to the 1930s when algebraic geometers understood that points of an algebraic variety "are" maximal (or prime) ideals of the ring of regular functions on it. The counterpart of this in analysis is the theory of commutative Banach algebras of Gelfand. Later Grothendieck revised the foundations of the whole algebraic geometry based on this idea, and it spread to many other areas of mathematics.

The idea goes back to the 1930s when algebraic geometers understood that points of an algebraic variety "are" maximal (or prime) ideals of the ring of regular functions on it. The counterpart of this in analysis is the theory of commutative Banach algebras of Gelfand. Later Grothendieck revised the foundations of the whole algebraic geometry based on this idea, and it spread to many other areas of mathematics.

To answer the comment: van der Waerden's Moderne Algebra, 1-st edition was published in 1930. In it a point (of a Riemann surface) is defined as a certain subring of the field (of meromorphic functions on this Riemann surface). And surely, van der Waerden is not the author of this idea: his book is based on lectures of Artin and Noether.