By David R. Finston and Patrick J. Morandi
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Extra info for An Introduction to Abstract Algebra via Applications
Ther operation can be considered as addition. However, we can introduce a new operations to come up with a ring. De…ne addition and multiplication on T by A + B = (A B) [ (B A B = A \ B: A) = (A [ B) (A \ B); Then R, together with these operations, forms a commutative ring. We leave the details of the proof to an exercise. We do point out some interesting properties of this ring. 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.
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. A useful alternative phrasing of the de…nition is the contrapositive. A nonzero element a 2 R is not a zero divisor if whenever b 2 R with a b = 0, then b = 0. Terminology about zero divisors varies from textbook to textbook. Some books only de…ne zero divisors for commutative rings. Some books consider 0 to be a zero divisor and others do not. Others talk about left and right zero divisors.
The hypothesis that a …eld contains at least two elements ensures that 0 6= 1; necessarily 0 = 1 if the …eld has only one element. If it has more than one, a short argument will prove that 0 6= 1. This assumption is not important, but it is convenient. It is similar to the assumption that 1 is not a prime number, even though it cannot be factored in a nontrivial way. 26. The number systems Q, R, and C are all examples of a …eld. The following proposition gives us ini…nitely many examples of …elds.