Is there some kind of a standard, uniform, or otherwise natural method for selecting representatives of classes of associate elements in certain rings? I would be interested in a simultaneous generalisation of positive integers and of unitary polynomials over a field, or of the absolute value function for integers and of the division of polynomials by the leading coefficient.
Is there a standard term for such an operation?
For example, when computing a PGCD of a family of integers, it is customary to look for the positive one, and when computing a PGCD of a family of univariate polynomials, it is customary to look for the unitary one.
(My current context: I am trying to program a generic function to compute the "normalised" PGCD, and I am interested in proper terminology and in any general theory on the subject.)