I'm writing code to do computation in algebraic number fields and am (re)-learning some algebra in the process.
When working with a ring, it seems useful to have an operation that "canonicalizes" a nonzero element with respect to multiplication by units. For example, in $\Bbb Z$, whose units are $\pm1$, we can take "canonical = positive". In the polynomial ring $K[X]$ where $K$ is a field, we can take "canonical = monic". This lets us (for example) produce unique representations for elements of a field of fractions by ensuring the denominators are canonical.
This got me wondering about other rings. In particular, in $\Bbb Z/n\Bbb Z$, is there a "natural" and/or efficient-to-compute canonicalization operation? Of course, we could just take the canonical unit multiple of a given element to be the smallest positive one (which will always be $\le n/2$ since $x\mapsto n-x$ is equivalent to multiplication by $-1$), but is there something "nicer" (maybe something that works in any quotient ring)? I realize this is an imprecise question; any insight is appreciated.
