6.2 (part 1), due February 25

Call me crazy, but when I hear "quotient" I think division and I don't see how these quotient rings have anything to do with division.  The notation looks like a fraction which is related to division, but I'm trying to ignore that because it's been throwing me off a little bit from the very beginning. So I guess my question is - is there any sort of division involved in quotient rings? The addition and multiplication of ideals is fairly straight forward when you pay close attention to the notation and what each bit means. As  I read through the section I almost instinctively foiled (a+I)(c+I), but then I realized that I^2 wouldn't make sense. You gotta be careful with this stuff.