Ideals of the ring of random variables We know that a random variable is just a measurable functions from a probability space $\Omega$ to the real numbers.
And that $0$ is one of them ($0^{-1} (-\infty,x)=X$ if $x\geq 0$ and $\varnothing$ otherwise), if $X,Y$ are random variables, then $X-Y$ and $XY$ are random variables. Since the sum and product is made in the usual sense, the operations are commutative, associative, and the product distributes over the sum. So the set of all random variables over $\Omega$, $\mathfrak{X}$ is a commutative ring with unity (Constants, and in particular $1$ are also random variables).
So, what can we say about the ideals of $\mathfrak{X}$? Do they have some particular form? Is this somehow noetherian? What can we say in a particular case, for example, $\Omega$ is a finite set with a very coarse $\sigma-$field?
 A: Don't know exactly if this is what you're looking for, so I'll throw my observations so far in hope of them being useful and correct.
For one thing, you know that if $I=( f_\alpha\,:\,\alpha\in A)$ and $1_{\alpha}(x)=\begin{cases}1&\text{if }f_\alpha(x)\ne 0\\ 0&\text{if }f_\alpha(x)=0\end{cases}$, then $I=(1_\alpha\,:\,\alpha\in A)$ 
This gives you somewhat a number of canonical sets of generators. And it tells you that:


*

*finitely generated ideals are generated by finitely many indicator functions

*with a bit more machinery, finitely generated ideals are principal: $$(1_{A_1},\cdots,1_{A_n})=\left(1_{A_1\cap\cdots\cap A_n}\right)$$

*if the $\sigma$ algebra is finite, the ring has principal ideals (it's never a domain, unless $\sigma=\{\emptyset,\Omega\}$). In fact, if the $\sigma$-algebra is finite, and $m$ is the cardinality of the finest partition of $\Omega$ in measurable subsets, then $\mathfrak X=\Bbb R^m$.

*the ring ceases to be noetherian as soon as you find a countable strictly ascending (or descending) chain in the $\sigma$-algebra.
