No they can't and shouldn't. But they have a right to expect you to take care of company business while you are on the clock, not play personal games on a company computer.
Opps, I forgot, you are a bastion of honesty. You said so yourself on this post.
For penitence you should get up off your butt, go to the men's room, look in the mirror and punch yourself in the nose. You expect no less from others!