My question arises from considering the following rings:
- The subring $R_1$ of $\mathbb{Q}$ with elements $\{a/b : b \space \text{odd}\}$
- The subring $R_2$ of $\mathbb{Q}$ given by $R_2 = \{\frac{a}{2^{k}3^{l}} | a \in \mathbb{Z}, k, l \in \mathbb{Z_{\ge 0}}\}$
- The subring $R_3$ of $\mathbb{Q}(x)$ of rational functions $P(x)/Q(x)$ such that $Q(0) \neq 0$ and $Q(1) \neq 1$
In all three cases it is fairly straightforward to find the group of units. I struggle however finding the irreducible elements and finding maximal and principle ideals in the given rings. Is there a general strategy to tackle such problems? What are things that I need to look out for/consider when dealing with such problems?
Thanks in advance