Was I wrong to check his email in this case? Was he being sketchy?

My boyfriend and I were doing long distance and it was okay but hehad a few borderline sketchy habits. He refused to put "in a relationship" on his facebook status, citing that the joke relationship he had had been there forever and was funny and he was attached to it, etc. However, when I asked him a question about this girl that kept messaging him he insisted they weren't even friends.
Later when I visited him, I saw a message in his inbox from her. She asked if he wanted to have dinner that night and he said "I can't, I'm doing an essay." While he had been doing an essay all day, the real reason is that I WAS VISITING... it seemed really sketchy, so I checked his email account, and found out they had hung out quite a bit. Alone. When I confronted my boyfriend about it, he got mad at me for snooping. I only did it because he was lying to me. Who is right?