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15 Rings and Polynomials fof iff -Sauron Ring theory has many applications in computing, counting, cryptography, and com- munications. Rings are sets with two operations, "+" and"."(usually called addition and multiplication, respectively). In many ways rings are like vector spaces, and much of our treatment of the theory of rings will mirror our treatment of vector spaces at the beginning of this book. The prototypical ring is the set Z with addi- tion and multiplication as the two operations. But there are many other interesting and useful examples of rings, including F[x], Mn(F), C(U;F), and B(X) for any normed linear space X. Ring theory shows that many results that are true for the integers also apply to these rings. We begin with a survey of the basic properties of rings, with a special focus on the similarities between rings and vector spaces. We then narrow our focus to a special class of rings called Euclidean domains, and to quotients of Euclidean

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