By L. E. Sigler (auth.)

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**Additional info for Algebra**

**Example text**

We prove a factorization theorem for functions and entitle it the fundamental morphism theorem for sets. Given any injection q: A ~ X there is defined a subset q(A) of X which is the range of Q. 4). Conversely, if we are given a set X and a subset A of X, then the identity function I A: A ~ X is an injection from A to X. 4 Definition. An injection q: A into X. ~ X is sometimes called an embedding of A Given any surjection cp:X as follows: ~ B there is defined a quotient set X/R of X 1. (x 1 Rx 2 if and only if cp(x 1 ) = cp(x 2 )) defines an equivalence relation R on X 2.

2. Let L be the following set of functions: {flf:IR-> IR and f(x) = ax a, b E IR, a # 0}. Which of the following statements are correct? + b for some (A) L together with composition is a group with neutral element /~>.. (B) L together with composition is not a group because g: IR -> IR such that g(x) = x 3 is a bijection and is not in L. (C) L together with composition is not a group because h:IR-> IR such that h(x) = 5 is not a member of L. (D) L together with composition is not a group because there exist functions f, gEL such that go f #fog.

E) None of the four alternatives completes a satisfactory sentence. EXERCISES 1. Prove that the product of two rings is itself a ring. 2. Show that no product ring of non trivial rings can be an integral domain. 3. Give an example of a ring without a unity. 4. Does the ring <&>X, domain? +, n, 0) have nontrivial divisors of zero? Is it an integral 5. On the set 7L x 7L we define the following two operations: (s, t) + (u, v) = (s + u, t + v) + tv, sv + (s, t) D (u, v) = (su tu). Show that <7L x 7L, +, 0, (0, 0)) is a ring.