After your company is successfully registered with Estonian Business Register, you will receive an email from the court with your company number and registration details.
The email is in Estonian and English with registration details PDF document in Estonian.
Email subject would look like this "Menetlustoimingust teavitamine, registriasi Ä xxxxxxx, Your company name OÜ (registrikood xxxxxxxx)".
Email content would look like this in English.
We hereby inform you that Ruling No. Ä xxxxxxxx concerning Your company name OÜ (registry code: xxxxxxxx) was adopted in the Registration Department of Tartu County Court on Month DD, YYYY. An uncertified transcript of the document is attached to this e-mail as a PDF file. Sincerely, Registration Department of Tartu County Court, Pikk 32, 44307, Rakvere
The registrikood or registry code is your company number and there is no other identificator used in Estonia.
You can also search for your company details from the Estonian Business Register as suggested here. You can print a PDF with your registration details in English for free.