Certainly some people talk about "integers" in an algebraic function field in the ring-theoretic sense -- algebraic functions that satisfy a monic polynomial over the field of rational functions in one variable. But I do not know how common this is.
For a translation of Emmy Noether's paper "Abstrakter Aufbau der Idealtheorie" I need a term for "ganzen Grossen" in an algebraic function field in that sense. This is the same term she uses for integers in an algebraic number field. When she wants to specify integers in the basic arithmetic sense, rational integers in algebraic number fields, she uses "ganzen Zahlen."
A word-by-word translation would make "ganzen Grossen" into "integer magnitudes." I have never heard anyone in English talk about "integer magnitudes" in algebraic number fields. I think everyone just says "integers." But is this also common in algebraic function fields?