In English we wouldn't say "the participation" , just participation is free
Elsewhere, "eine Teilnahme ist kostenlos" is translated as "admission is free," but DL will not accept "the admission is free" here. Does using the definite versus indefinite article change the meaning of "Teilnahme?"
I think the problem is with the nouns, not the articles. "Admission" = Eintritt, while "Teilnahme" = Participation. They are similar, but not synonymous. If you're going to the theatre, it would always be admission/Eintritt, never participation/Teilnahme.
That would be "Der Teilnahmer ist kostenlos." (die Teilnahme = participation, but participant = der Teilnahmer)