By David R. Finston and Patrick J. Morandi

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**Example text**

It is possible that the ring properties are veri…ed in a calculus class. We do not verify them all here, but discuss some. The additive identity of R is the zero function z, which is de…ned by z(x) = 0 for all x 2 [0; 1]. The multiplicative identity of R is the constant function 1 de…ned by 1(x) = 1 for all x 2 [0; 1]. Commutativity of addition holds because if f; g 2 R and x 2 [0; 1], then (f + g)(x) = f (x) + g(x) = g(x) + f (x) = (g + f )(x): Thus, the functions f + g and g + f agree at every value in the domain, which means, by de…nition, that f + g = g + f .

40 CHAPTER 2. 5 The Golay Code In this section we discuss an example of a code which is called the extended Golay code. This code is the set of solutions to the matrix equation Hx = 0, where H is the 12 24 matrix whose left half is the 12 12 identity matrix and whose right half is 1 0 1 1 0 1 1 1 0 0 0 1 0 1 B 1 0 1 1 1 0 0 0 1 0 1 1 C C B B 0 1 1 1 0 0 0 1 0 1 1 1 C C B C B B 1 1 1 0 0 0 1 0 1 1 0 1 C C B B 1 1 0 0 0 1 0 1 1 0 1 1 C C B B 1 0 0 0 1 0 1 1 0 1 1 1 C C: B B=B C 0 0 0 1 0 1 1 0 1 1 1 1 C B B 0 0 1 0 1 1 0 1 1 1 0 1 C C B C B B 0 1 0 1 1 0 1 1 1 0 0 1 C C B B 1 0 1 1 0 1 1 1 0 0 0 1 C C B @ 0 1 1 0 1 1 1 0 0 0 1 1 A 1 1 1 1 1 1 1 1 1 1 1 0 This length 24 code was used by the Voyager spacecraft to photograph Jupiter and Saturn between 1979 and 1981.

However, we 50 CHAPTER 3. RINGS AND FIELDS have seen that Zn does not always share this property. For example, in Z6 we have 2 3 = 0. Likewise, this bad behavior occurs with matrices. For example, 1 0 0 0 0 0 0 1 0 0 0 0 = : If we are trying to solve equations in a ring, then it may be crucial to know whether or not we can have the product of nonzero elements equal to zero. 17. If R is a ring, then a nonzero element a 2 R is said to be a zero divisor if there is a nonzero element b 2 R with a b = 0.

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