# What are some applications of Principal Ideal Rings

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.

• The polynomial ring in one variable over a field. – Wuestenfux Nov 6 '17 at 19:58

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)
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 $T$.
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).