[jvvw]

Just as a warning most maths algebra beginner textbooks will focus on groups, rings and fields and not talk about most of the other structures mentioned (including Fragleigh and Herstein recommended by others - they are incredibly different styles, and although I have extremely fond memories of Herstein, there are better books than those two especially if you aren't actually doing a maths degree, but my knowledge of textbooks is 20 years out of date). On the other hand once you have a good grasp of a few structures, learning about and understanding others becomes much easier. I guess it all depends on why you want to learn about them and what you want to do with them