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The investigation of the nature of division rings and the attempts to classify them form an important part of algebra. 3.2 Some Special Classes of Rings. The In this lecture, we start with a pseudo-historical note on algebra. Next ring is defined and some examples are briefly mentioned. Ring of polynomials and algebra, as Hamilton called it, launched non-commutative ring theory. Other natural non-commutative objects that arise are matrices. They were.Course notes for MATH3002 Rings and Fields Chapter 5: Some Ring Theory Abstract algebra is essentially a tool for other branches of mathematics.