Quoting the CRO : "Registration of a business name is obligatory if any individual or partnership (whether composed of individuals or bodies corporate or any combination of both) or any body corporate carries on business under a name other than their own true names. "
"Specifically registration of a business name is required if: -an individual uses a business name which differs in any way from his/her true surname. It makes no difference whether the individuals first name or initials are added. So registration is required if, for example, Mr. John Murphy traded as Murphy Builders but not if he traded as Murphy or John Murphy); -a firm uses a business name which differs in any way from the true names of all partners who are individuals and the corporate names of all partners which are bodies corporate; -a company uses a business name which differs in any way from its full corporate name; -a person having a place of business in the State carries on the business of publishing a newspaper."
For more help with your .ie domain application, you can see the below help articles: