By David R. Finston and Patrick J. Morandi

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**Example text**

First of all, 0 = ? for this ring since A + ? ) = A ? = A for any subset A of T . Also, 1 = T since A T = A \ T = A for any subset A. Next, for any A we have A + A = (A [ A) (A \ A) = A A = ?. Therefore, A = A for any A! This would seem to be a very unusual property, although we will see it all the time when we deal with coding theory. Finally, note that A A = A \ A = A. This example is important in the study of set theory and logic. 12. In this example we describe a method of constructing a new ring from two existing rings.

That is, given any two subsets of T , the union and intersection of the sets is again a subset of T . 4. Here is an example from multivariable calculus. Consider the set R3 of 3tuples of real numbers. The cross product is an operation on R3 . Recall that the cross product is given by the formula 0 1 i j k (a1 ; a2 ; a3 ) (b1 ; b2 ; b3 ) = det @ a1 a2 a3 A b1 b2 b3 = (a2 b3 a3 b2 )i + (a3 b1 a1 b3 )j + (a1 b2 + a2 b1 )k; Here i, j, and k are the usual unit vectors (1; 0; 0), (0; 1; 0), and (0; 0; 1).

For, if r 2 R and s 2 S, then (r; 0) (0; s) = (0; 0). For example, if R = S = Z, then elements of the form (n; 0) or (0; m) with n; m nonzero are zero divisors in Z Z. There is a cancellation law of multiplication for the ordinary number systems. Because of the existence of zero divisors, the generalization to arbitrary rings is more complicated. 22 (Cancellation Law of Multiplication). Let R be a ring. Suppose that a; b; c 2 R with ab = ac and a 6= 0. If a is not a zero divisor, then b = c. Proof.

### An Introduction to Abstract Algebra via Applications by David R. Finston and Patrick J. Morandi

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