I'm trying to understand ideals in rings, more particularly in the ring $\mathbb{Z}$. I'm looking at the ideals $(4,6),(4)\cap(6)$ and $(4)+(6)$. Now my question is, knowing that $\mathbb{Z}[I]$ is a PID and a UFD, what is the "easiest" way of finding generators for these ideals.
I might be wrong, but I think that $(4)+(6)=(\text{gcd}(4,6))=(2)$ and $(4)\cap(6)=(\text{lcm}(4,6))=(12)$. Now what about $(4,6)$, how would you find a generator for this ideal?
Any help is greatly appreciated.
Welcome to MSE!
Let me answer the question implicitly underlying this one, which is "how do we understand (read: compute with) ideals?".
Ideals generalize divisibility, in the sense made precise by this slogan (which you should commit to memory):
$$\large \text{To contain is to divide}.$$
That is, we should think of the containment $I \supseteq J$ as being analogous to $I \mid J$. Note the direction! $I$ contains $J$, thus $I$ "divides" $J$. To contain is to divide.
Now, for principal ideals, this analogy is literally true! $(r) \supseteq (s) \iff r \mid s \ $ (if this isn't obvious, it's worth proving to yourself). In particular, for PIDs, this tells you how to do any computation you might want to do with ideals. This might not be immediately obvious, but it becomes obvious when you realize how deep the above slogan goes.
Indeed the divisibility lattice for our ring $R$ is isomorphic to the lattice of principal ideals of $R$ turned upside down!
So the meet of two principal ideals $(r) \cap (s)$ is the join of those elements in the divisibility lattice. That is, the lcm of $r$ and $s$ (as an aside, do you see why this is the join in the divisibility lattice?)
Similarly, the join of two principal ideals $(r,s) = (r) + (s)$ (note this is the smallest ideal containing $r$ and $s$, so it really is the join) is exactly the meet of $r$ and $s$ in the divisibility lattice. That is, the gcd of $r$ and $s$.
Indeed, historically, the reason people studied ideals at all was because they gave "idealized" elements that make divisibility work better! The nonprincipal ideals (of dedekind domains, say) fill in elements of a divisibility lattice that don't really exist, but which give a much nicer theory. I'll leave a broader discussion of this phenomenon to either a book on the history of mathematics (for instance, chapter 27 of Stillwell's excellent Mathematics and its History) or on algebraic number theory (for instance, chapter 3 of Stein's excellent Algebraic Number Theory: A Computational Approach, or many other wonderful books)
Now for the question that you actually asked!
$(4,6)$ is the join of $(4)$ and $(6)$, it's principal (since every ideal of $\mathbb{Z}$ is) so it's generated by the meet of $4$ and $6$ in the divisibility lattice of $\mathbb{Z}$. That is, the gcd $2$. So $(4,6) = (2)$.
As another example, one can ask about $(2) \cap (i)$ in the gaussian integers $\mathbb{Z}[i]$. This is the meet, so if it's principal (which it is) it should be generated by the join of $2$ and $i$. That is, by the lcm $2i$. So $(2) \cap (i) = (2i)$.
Lastly, this also lets us make computations involving products of ideals. For instance, in $\mathbb{Z}$, we have
$$ (2)(4) = (8) $$
Since the lcm always divides the product, do you see why the above discussion automatically tells you that $I \cap J \supseteq IJ$?
As another instance of the utility of this analogy for making computations, say that $I$ and $J$ are comaximal in the sense that $I+J = (1)$.
What does this tell us about the "gcd" of $I$ and $J$? Do you see how one might guess that, in this case, $I \cap J = IJ$? This turns out to be true.
I hope this helps ^_^
