Is there an elementary way to see that a product of an infinite number of non-units in a (noncommutative) ring is always $0$? Here, by non-units I mean elements that are not left-invertible.
This statement should be equivalent to every principal left ideal being Noetherian. Since the latter is true for any ring, the former must also be true, but I couldn't find an elementary proof.
It is not true, even if you have a topological ring, and even if you assume the infinite product exists.
For example, the ring: $$\mathbb Z[1/2]=\left\{\frac a{2^k}\mid a,k\in\mathbb Z, k\geq 0\right\}$$ with the topology inherited from the rational numbers.
The units are $\pm 2^j$ for any $j\in\mathbb Z.$
Then the product: $$\frac{3}{4}\cdot \frac{5}{4}\cdot \frac{17}{16}\cdots \frac{2^{2^k}+1}{2^{2^k}}\cdots$$
converges to $1,$ but all of the terms are non-units. (The first term, $\frac34,$ does not match the general pattern, but the other terms do.)
A partial product, up to the term for $k,$ can be shown to be $$\frac{2^{2^{k+1}}-1}{2^{2^{k+1}}}$$ by induction.