Is it appropriate to put your bachelor degree in the signatures of your emails?

I don't think its INappropriate. But it would certainly be viewed as tacky. Especially in the business world, a Bachelor's isn't a big deal. Its the standard for education. The reason folks list a PhD or Masters is that those are above and beyond the standard. I think that's is still a little pretentious, but I recognize the work they've done and I'll give them that bit of recognition. To list your BA will still have the pretentiousness WITHOUT the extra work. You'll look silly.