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What are some applications of principal ideal rings? I have searched google and scholar but to no avail. It would be helpful if you can point me in the right direction.

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    $\begingroup$ The polynomial ring in one variable over a field. $\endgroup$ – Wuestenfux Nov 6 '17 at 19:58
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Every cyclic linear code of length $n$ over the field $F$ is a quotient ring of the form $F[x]/(x^n-1)$. This allows you to completely classify all the generator and parity-check polynomials for these codes, and enables you to make some guarantees about minimum distance in such codes. (See for example BCH codes)

Discrete valuation rings are also a class of PIR that get a lot of mileage.

Finally, much of elementary linear algebra can be viewed through the lens of the PIR $F[T]\cong F[x]/(p(x))$ where $p(x)$ is the minimal polynomial for a linear transformation $T$.

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A number-theoretic application is the study of Diophantine equations, e.g., of $x^n+y^n=z^n$. Euler proved for $n=3$ that there are no non-trivial integral solutions. He used that the ring of integers $\mathbb{Z}(\zeta_3)$ of the cyclotomic field $\mathbb{Q}(\zeta_3)$ is a PID (Principal Ideal Domain). Unfortunately, $\mathbb{Z}(\zeta_p)$ is a PID for primes $p$ if and only if $p\le 19$. So Fermat cannot be solved this way (this was, what Kummer recognised).

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