Soda ash ='s sodium carbonate.
https://www.etisoda.com/en/what-is-soda-ash/
https://en.wikipedia.org/wiki/Sodium_carbonate
Depending upon source American publications, media, persons, etc, go back and forth using either "soda ash" or "sodium carbonate".
It's strange because within same publication you'll find references to "soda ash" for making eau de Javel (chlorine bleach). Elsewhere "sodium carbonate" is used for say making built soap, as builder for wash water when doing laundry and so on.
Regarding soaps for wash day, without much blather what's wanted is either high or low titer soap.
High titer soaps that work best at high or boiling wash temperatures.
Low titer are soaps that are more gentle and used in warm or cooler water.
Fels, Duz, Oxydol many other commonly known soap products from back day were high titer. OTOH Ivory Flakes/Snow, Lux Flakes and similar products meant for fine washables were usually low titer soaps.
Perhaps most famous soap today made from plant ash is France's "Savon de Marseille".
Used for everything from washing laundry to little faces for over 600 years ingredients of Savon de Marseille have been regulated by French law.