Exhibit an integral domain $R$ and a non-zero non-unit element of $R$ that is not a product of irreducibles. 
Exhibit an integral domain $R$ and a non-zero non-unit element of $R$ that is not a product of irreducibles.

My thoughts so far: I don't really have a clue. Could anyone direct me on how to think about this? I'm struggling to get my head round irreducibles.
Thanks.

Such an element can be factored, each factor can be factored, each factor can be factored, etc.
Changing the problem into an additive one, you would want to find an element that can be written as a sum of two strictly smaller numbers, each of which can be written as a sum of two strictly smaller numbers, each of which... etc.
Perhaps thinking along the lines of:
$$1 = \frac{1}{2}+\frac{1}{2} = \left(\frac{1}{4}+\frac{1}{4}\right) + \left(\frac{1}{4}+\frac{1}{4}\right) = \cdots = \left(\frac{1}{2^n}+\frac{1}{2^n}\right) + \cdots + \left(\frac{1}{2^n}+\frac{1}{2^n}\right) = \cdots$$
Hmmm... Is there any way we could turn that into some kind of multiplicative, instead of additive, set of equalities? 
