Let S = { la, b e R} and let p : s –R be defined by : ø( ) = c 1) is a ring a) Homomorphism. b) Isomorphism c) None 2) ker (o) is generated by a) o b) C ] c) 3) S/ker () a) 2 b) c) d) None

. a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show that is not a field. Hence Theorem is not true if the condition that is commutative is removed. Theorem 6.22 Quotient Rings That are Fields. Let be a commutative ring with unity, and let be an ideal of . Then is a field if and only if is a maximal ideal of .

22. Let be a ring with finite number of elements. Show that the characteristic of divides .

29. Let be the set of Gaussian integers . Let . a. Prove or disprove that is a substring of . b. Prove or disprove that is an ideal of .

11. Show that defined by is not a homomorphism.

Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .

18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .

Let be the ring of Gaussian integers. Let divides and divides. Show that is an idea of. Show that is a maximal ideal of.

14. Let be an ideal in a ring with unity . Prove that if then .

[Type here] Examples 5 and 6 of Section 5.1 showed that is a commutative ring with unity. In Exercises 4 and 5, let . 4. Is an integral domain? If not, find all zero divisors in . [Type here]