John R. Durbin
This text is appropriate for any one-semester junior/senior level course in Modern Algebra, Abstract Algebra, Algebraic Structures, or Groups, Rings and Fields. Durbin has two main goals: to introduce the most important kinds of algebraic structures, and to help students improve their ability to understand and work with abstract ideas. The first six chapters present the core of the subject; the remainder are designed to be as flexible as possible. Durbin covers groups before rings, which is a matter of personal preference for instructors. The course is mostly comprised of mathematics majors, but you will find engineering and computer science majors as well.
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Modern Algebra: An Introduction