Bellerin is on a season-long loan with Betis.
Arteta said: "He's been here a long time and he's been a really important part of everything that happened at the club in the last 10 years.
"He's an exceptional player but an incredible person, and he will be missed. To find somebody like Hector is not easy. He's been raised here with us and is loved by everyone at the football club.
"I have to say goodbye with sadness but at the same time, I had to understand his position. He needed a new challenge. He's needed a new challenge for a while. After discussing it for a long time, we decided to let him go because we believe it was the right thing for him as well."