How Do Radical Ideals Relate to Submodules in Algebra?

[alexfloo]

I'm currently studying commutative algebra/algebraic geometry out of Cox Little and O'Shea's Ideal Varieties and Algorithms, and linear algebra out of Steven Roman's Advanced Linear Algebra. In Roman, I'm learning about modules, and I have a question about the relationship between these two fields.

An ideal is exactly a submodule of F[x₁, ..., x_n] as a module over itself. We know that an ideal over this set exactly determines a subset of the affine space Fⁿ. On the other hand, affine subsets of Fⁿ define radical ideals of F[x,y] which are in turn submodules.

Now outwardly, the property of being radical depends on the multiplicative structure of the ideal, which the corresponding submodule doesn't have. My question is whether the property of being radical depends innately on that multiplicative structure. Do submodules corresponding to radical ideals have any identifiable properties even without recognizing their multiplicative structure?

(Wikipedia tells me that there is such a thing as a radical submodule, but I don't think it's what I'm looking for.)

 

[fresh_42]

This is the reason why modules and ideals are two different things. The correspondence is only a module isomorphisms, since modules do not carry a ring structure themselves. You cannot transport the definition of a radical ideal into the module language, since multiplication is missing. But this is already true for arbitrary ideals.