§ General operations on ideals

We have at our hands a commutative ring $R$, and we wish to study the ideal structure on the ring. In particular, we can combine ideals in the following ways:

1. $I + J \equiv \{ i + j : \forall i \in I, j \in J \}$
2. $I \cap J \equiv \{ x : \forall x \in I \land x \in J \}$
3. $I\oplus J \equiv \{ (i, j) : \forall i \in I \land j \in J \}$
4. $IJ \equiv \{ ij : \forall i \in I \land j \in J \}$

We have the containment:

§ $IJ$ is a ideal, $IJ \subseteq I \cap J$

it's not immediate from the definition that $IJ$ is an ideal. The idea is that given a sum $\sum_k i_k j_k \in IJ$, we can write each $i_k j_k = i'_k$, since the ideal $I$ is closed under multiplication with $R$. This gives us $\sum i'_k = i'' \in I$. Similarly, we can interpret $\sum_k i_k j_k = \sum_k j'_k = j''k \in J$.

Hence, we get the containment $IJ \subseteq I \cap J$.

§ $I \cap J subseteq I$, $I \cap J \subseteq J$

Immediate from the inclusion function.

§ $I, J \subseteq I + J$

Immediate from inclusion

§ CRT from an exact sequence

There exists an exact sequence:

We are forced into this formula by considerations of dimension. We know:

By analogy to euler characteristic which arises from homology, we need to have $I \oplus J$ in the middle of our exact sequence. So we must have:

Now we need to decide on the relative ordering between $I \cap J$ and $I + J$.

• There is no universal way to send $I oplus J \rightarrow I \cap J$. It's an unnatural operation to restrict the direct sum into the intersection.
• There is a universal way to send $I \oplus J \rightarrow I + J$: sum the two components. This can be seen as currying the addition operation.

Thus, the exact sequence must have $I + J$ in the image of $I \oplus J$. This forces us to arrive at:

The product ideal $IJ$ plays no role, since it's not possible to define a product of modules in general (just as it is not possible to define a product of vector spaces). Thus, the exact sequence better involve module related operations. We can now recover CRT:

§ References

• I learnt the material from this course on commutative algebra from IIT bombay .