Proving a subring of $\Bbb Z[\zeta_{11}]$ is PID

Firstly, the discriminant of $K = \mathbb{Q}[\zeta_{11}+\zeta_{11}^{-1}]$ is $11^4$, so Minkowski's bound is $\leq 4$.

However, both primes $2$ and $3$ are inert in $K$, so there is no ideal of norm $2, 3$ or $4$. Therefore the class group is trivial.

In the same way, one can prove, without any calculation, $\mathbb{Z}[\zeta_p+\zeta_p^{-1}]$ is a PID when $p=13$; while $p=17,19$ require some computations and clever ideas.

Your question is equivalent to show that the class number (= the order of the ideal class group) of the field $\mathbf Q(\zeta_{11}+\zeta_{11}^{-1})$ is $1$. More generally, denote $K_m=\mathbf Q(\zeta_m), K_m^+$ = the maximal totally real subfield of $K_m$, $h_m$ and $h_m^+$ their respective class numbers. It is a classical result that $h_m^+$ divides $h_m$; write $h_m=h_m^+ h_m^-$. The "minus" class number $h_m^-$ may be considered as known, thanks to an analytical formula giving it in terms of Bernoulli numbers (see e.g. Washington, "Introduction to cyclotomic fields", thm. $4.17$). The "plus" class number $h_m^+$ is much more difficult to compute. Around $1975$, using deep results of Odlyzko on lower bounds of discriminants of number fields, Masley was able to determine all the $K_m$ with $m\neq 2$ mod $4$ for which $h_m=1$. A complete (finite) list is given in op. cit., thm. $11.1$. In particular, for an odd prime $p$, it appears in that list that $h_p =1$ for all $p\neq 23,29,31,37$. "Horrendeous" ad hoc calculations show e.g. that $h_{23}^+=1$, see https://math.stackexchange.com/a/90005/300700 and https://math.stackexchange.com/a/2024190/300700, but the problem of the determination of all $p$ for which $h_p^+=1$ remains unsolved in general. To have an idea of the computational difficulties, see e.g. R. Schoof's paper "Class Numbers of Real Cyclotomic Fields of Prime Conductor" (2002), in which the author stresses that " the class numbers $h_p^+$ are notoriously hard to compute. Indeed, the number $h_p^+$ is not known for a single prime p ≥ 71."