The interpretation of ideals of a ring.

Question

Ideals of a commutative ring (I have only studied the commutative case) are thought of as generalized numbers (in algebraic number theory) and as ring homomorphisms (through the ideal as kernel formulation). Is there some way of identifying these two interpretations? Maybe one could identify the product and sum operations on ideals with some operations on the homomorphisms?

One might have to restrict the rings we are looking at further to get some meaningful interpretation but that is fine.

Answer 1

Ideals are not generalized numbers
It is true that ideals were introduced by Dedekind as "ideal numbers" in order to restore unique factorization in certain rings where it fails.
These rings are now called Dedekind rings and indeed it is a fruitful point of view to treat ideals in these rings as generalized numbers.
However these rings have Krull dimension $1$ and are very special.
It is completely impossible to treat ideals in general rings as numbers: for example it makes no sense at all to say that the ideal $I=\langle x^2+y^2+z^2, x^3+y^3+z^3\rangle\subset \mathbb C[x,y,z]$ has anything to do with a generalized number.

Ideals are varieties !
In the 1950's Grothendieck introduced a wonderful point of view in algebraic geometry allowing us to consider any commutative ring $A$ as a geometric entity called the affine scheme $\operatorname {Spec}(A)$.
It generalizes the classical algebraic varieties and in this vision the ideals $I\subset A$ correspond exactly to the subobjects, called subschemes, $V(I)\subset \operatorname {Spec}(A)$ .

The sum of ideals exactly corresponds to the intersection of subschemes: $V(I+J)=V(I)\cap V(J)$ and the product of ideals exactly corresponds to the union of subschemes: $V(I\cap J)=V(I)\cup V(J)$.
The product $I\cdot J$ of ideals is a bit more difficult to interpret and should be thought of as yielding some approximation of the union of subschemes: $V(I\cdot J)\cong V(I)\cup V(J)$

Finally in the example I evoked above $ \operatorname {Spec}(\mathbb C[x,y,z])$ should be thought about as the vector space $\mathbb C^3$ and $V(I)$ as the curve in that vector space defined by the equations $x^2+y^2+z^2=0$, $x^3+y^3+z^3=0$ .

Answer 2

The sum of ideals, at least, has a categorical interpretation: It corresponds to the pushout of the epimorphisms that the two ideals being added corespond to.

You have a ring $R$ with ideals $\mathfrak I$ and $\mathfrak J$ and canonical homomorphisms $\pi_1:R\to R/\mathfrak I$ and $\pi_2:R\to R/\mathfrak J$.

Then whenever you have a homomorphism $f:R\to S$ which factors through $\pi_1$ and $\pi_2$ separately (that is, $f=g_1\circ\pi_1$ and $f=g_2\circ\pi_2$), then $f$ also factors uniquely through the canonical homomorphism $R\to R/(\mathfrak I+\mathfrak J)$. And this characterizes $R/(\mathfrak I+\mathfrak J)$ up to isomorphism.

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(Of course, however, you should beware that this doesn't quite connect to "ideals as generalized numbers": The sum of principal ideals is not the principal ideal generated by the sum of the generators).