Modules are just much more flexible than ideals. Two major advantages:
Richer geometry. An ideal is a closed subscheme of Spec(R), while modules are quasicoherent sheaves. An element x of M is a global section of the associated sheaf, and the ideal Ann(x) corresponds to the vanishing locus of that section. This leads to a nice geometric picture of associated primes and primary decomposition which explains how finitely generated modules are built out of modules R/P with P prime ideal (I am not an algebraist at heart, so for me the only way to remember the statement of primary decomposition is to translate from geometry 😅)
Richer (homological) algebra. Modules form an abelian category in which ideals do not play an especially prominent role (unless one looks at monoidal structure but let’s not go there). The corresponding homological algebra (coherent sheaf cohomology, derived categories) is the core engine of modern algebraic geometry.
Historically commutative algebra came out of algebraic number theory, and the rings involved—Z,Z_p, number rings, p-adic local rings… - are all (in the modern terminology) Dedekind domains.
Dedekind domains are not always principal, and this was the reason why mathematicians started studying ideals in the first place. However, the structure of finitely generated modules over Dedekind domains is still essentially determined by ideals (or rather fractional ideals), reflecting to some degree the fact that their geometry is simple (1-dim regular Noetherian domains).
This could explain why there was a period where ring theory developed around ideals but the need for modules was not yet clarified?
Modules are just much more flexible than ideals. Two major advantages:
Richer geometry. An ideal is a closed subscheme of Spec(R), while modules are quasicoherent sheaves. An element x of M is a global section of the associated sheaf, and the ideal Ann(x) corresponds to the vanishing locus of that section. This leads to a nice geometric picture of associated primes and primary decomposition which explains how finitely generated modules are built out of modules R/P with P prime ideal (I am not an algebraist at heart, so for me the only way to remember the statement of primary decomposition is to translate from geometry 😅)
Richer (homological) algebra. Modules form an abelian category in which ideals do not play an especially prominent role (unless one looks at monoidal structure but let’s not go there). The corresponding homological algebra (coherent sheaf cohomology, derived categories) is the core engine of modern algebraic geometry.
Historically commutative algebra came out of algebraic number theory, and the rings involved—Z,Z_p, number rings, p-adic local rings… - are all (in the modern terminology) Dedekind domains.
Dedekind domains are not always principal, and this was the reason why mathematicians started studying ideals in the first place. However, the structure of finitely generated modules over Dedekind domains is still essentially determined by ideals (or rather fractional ideals), reflecting to some degree the fact that their geometry is simple (1-dim regular Noetherian domains).
This could explain why there was a period where ring theory developed around ideals but the need for modules was not yet clarified?