# John E. Maxfield's Abstract Algebra and Solution by Radicals PDF

By John E. Maxfield

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**Sample text**

Definition 2-3. A commutative ring is a ring in which multiplication is commutative. Definition 2-4. An integral domain is a commutative ring with unit element with no proper divisors of zero; that is, a product a • b cannot equal zero unless one of the factors ¿7 or is zero. 20 O T H E R A B S T R A C T A LG E B R A S Definition 2-5. ” These definitions are more restrictive than Definition 1-2 of groups; that is, they impose more than the group requirements. One abstract algebra with fewer restrictions than a group is the semigroup.

How are {a, b) and (c, cl) related if they have the same solution s i bs = a implies ci{bs) = d(a), so that bds = ad, and ds- = c implies b{ds) = 6 (c), so that bds = 6c, so that we have ad = be. Exercise 2-15. Write four number pairs equivalent to (4, 1). Exercise 2-16. Prove that if we write (a, 6) ^ (c, d) whenever ad = be, then ~ is an equivalence relation on ordered pairs (x ,y ) with x ,y e l and y 0. Take it as known that / under multiplication is associative and com mutative and has the cancellation property ac = be implies a = 6 for any c t*£ 0.

See Definition 1-2 ii). One way of adjoining an additive identity to N is to lump all these problems together in an “equivalence class” and equate the answer to the problems: Dropping all but the pair of numbers (