Is it OK to write Emmy Nöther instead of Emmy Noether? I always assumed that the oe was an anglicization of ö (o with an umlaut). But when I look her up, I never see Nöther.
Taking a look at her 1933 paper in Mathematische Annalen one sees:
Similarly for a 1923 paper:
From a glance at a few other papers, she (or all the journals) used "Noether" for her last name. Further, since the titles clearly indicate that the journals are quite happy with umlauts it is not by accident.
As you can read on the Wikipedia article about Emmy's father, Max Noether, the family only received their name in 1809. The generation of the great-grandfather of Emmy.
The first spelling I found of Elias Samuel, is "Netter", but "Noether" was already used as alternative: https://www.geni.com/people/Elias-Noether/6000000006433734670
His son, Hermann (Herz Elias) Noether seems to have used "Nöther" as a surname. https://www.geni.com/people/Hermann-Herz-N%C3%B6ther/6000000002955209487, it's also spelled Nöther on his tombstone: https://images.findagrave.com/photos/2021/141/138798362_02a7afc7-e049-4bf2-9185-973df3f39bf9.jpeg
From Emmy's father, Max, onwards, the family seems to standardise on "Noether", perhaps because they already had more international contacts: https://www.geni.com/people/Max-Noether/6000000002955209501
Even on German signs, Emmy's surname is spelled as Noether. So it's safe to assume that this was the preferred spelling by the time Emmy lived. Though other spellings have been used in their family.