Reading "H. Matsumura - Commutative Ring theory" I had the impression that the definitions were all made to mean something in algebraic geometry afterwards. I wonder what was commutative algebra before the modern algebraic geometry (which ideas were developed, how sophisticated they were, etc). Moreover, whether the success of algebraic geometry had inspired the expansion of commutative algebra, and how much. Any older reference would also be interesting (not for studying, just to figure out). Thanks
here is an excerpt from Zariski's talk at the Icm several decades ago:
"The arithmetic trend in algebraic geometry is not in itself a radical departure from the past. This trend goes back to Dedekind and Weber who have developed, in their classical memoir, an arithmetic theory of fields of algebraic functions of one variable. Abstract algebraic geometry is a direct continuation of the work of Dedekind and Weber, except that our chief object is the study of fields of algebraic functions of more than one variable. The work of Dedekind and Weber has been greatly facilitated by the previous development of classical ideal theory. Similarly, modern algebraic geometry has become a reality partly because of the previous development of the general theory of ideals. But here the similarity ends. Classical ideal theory strikes at the very core of the theory of algebraic functions of one variable, and there is in fact a striking parallelism between this theory and the theory of algebraic numbers. On the other hand, the general theory of ideals strikes at most of the foundations of algebraic geometry and falls short of the deeper problems which we face in the post- foundational stage. Furthermore, there is nothing in modern commutative algebra that can be regarded even remotely as a development parallel to the theory of algebraic function fields of more than one variable. This theory is after all itself a chapter of algebra, but it is a chapter about which modern algebraists knew very little. All our knowledge here comes from geometry. For all these reasons, it is undeniably true that the arithmetization of algebraic geometry represents a substantial advance of algebra itself. In helping geometry, modern algebra is helping itself above all. We maintain that abstract algebraic geometry is one of the best things that happened to commutative algebra in a long time."