Hint $\ $ An ideal $\rm\,I\neq 0\,$ in a PID is generated by any $\rm\,0\neq b\in I\,$ with least number of prime factors, since such minimality implies that $\rm\,b\,$ divides every $\rm\,c\in I\ $ (else $\rm\ d = gcd(b,c)\mid b\ $ properly, $ $ therefore $\,\rm d\,$ has fewer primes factors than $\rm b,\,$ and $\rm\, (d) = (b,c) \subset I,\,$ contra minimality of $\rm b).$
In your PID $\rm\,D,\,$ all odd primes $\rm\,p\,$ are units by $\rm\,1/p \in D.\,$ So the only prime that survives in $\rm\,D\,$ is $\rm\,p=2.\,$ Thus, by above, an ideal of $\rm\,D\,$ is generated by any one of its elements having the least number of factors of $\,2.\,$ Thus every ideal has form $\rm\,(2^n),\,$ hence every ideal is principal.
Remark $\,$ Implicit in the above is the following pretty generalization of the Euclidean algorithm to arbitrary PIDs. The Dedekind-Hasse criterion states that a domain $\rm\,D\,$ is a PID iff given any $\rm\,0\ne b,c \in D,\,$ either $\rm\,b\mid c\,$ or there exists a $\rm D$-linear combination of $\rm\,b,c\,$ smaller than $\rm b,\,$ where size is measured by naturals (or any ordinal), so that induction (or descent) works.
It is clear that such a domain must be a PID, since the smallest element in an ideal must divide all others. Conversely, since a PID is UFD, an adequate metric is the number of prime factors (since if $\rm\,b\nmid c\,$ then their gcd $\rm\,d\,$ must have fewer prime factors; for if $\rm\,(b,c) = (d)\,$ then $\rm\,d\,|\,b\,$ properly, else $\rm\,b\,|\,d\,|\,c\,$ contra hypothesis). Notice Euclidean descent by the Division Algorithm is just a special case, hence Euclidean $\Rightarrow$ PID ($\Rightarrow$ {UFD, Bezout} $\Rightarrow$ GCD).