Why do we say "to invite someone for _ dinner" without any article whereas in "to go for a walk" we use one?
Is there any general rule covering the cases or should they just be remembered?
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It's just idiomatic:
invite someone to dinner and all the other meals, too. [breakfast,lunch, supper and tea]
You can use for, also. That is slightly different than: I was invited to a dinner when I was in Paris. The dinner was very formal. However, specifically, you might say a instead of nothing: a dinner, a lunch when explaining it. Tea is a bit different. No "a tea" here.
Movement and transportation:
go for a drive, for a walk, for a spin, for a run, for a swim, for a ride (horses or bikes or motorbikes) = as a generic idea for an activity
The phrases works as one, though drives and walks are often used in the plural, too.