In addition to being a beautiful and deep theory in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytic geometry. Let us start with a historical survey of its development.

The most basic commutative rings are the ring ℤ of rational integers, and the polynomial rings over a field. ℤ is a principal ideal ring, and so is too simple to be ring-theoretically very interesting, but it was in the course of studying its extensions, the rings of integers of algebraic number fields, that Dedekind first introduced the notion of an ideal in the 1870s. For it was realised that only when prime ideals are used in place of prime numbers do we obtain the natural generalisation of the number theory of ℤ.

Meanwhile, in the second half of the 19th century, polynomial rings gradually came to be studied both from the point of view of algebraic geometry and of invariant theory. In his famous papers of the 1890s on invariants, Hilbert proved that ideals in polynomial rings are finitely generated, as well as other fundamental theorems. After the turn of the present century had seen the deep researches of Lasker and Macaulay on primary decomposition of polynomial ideals came the advent of the age of abstract algebra. A forerunner of the abstract treatment of commutative ring theory was the Japanese Shōzō Sono (On congruences, I–IV, Mem. Coll Sci. Kyoto, 2 (1917), 3(1918–19)); in particular he succeeded in giving an axiomatic characterisation of Dedekind rings.