Denote by Zm = {[0]m, [1]m, ..., [m1]m} the ring of integers modulo m. Consider the rings R = Z24 and SZ4 × Z6. Let 0 RS be the map defined by ([x]24) = ([x]4, [4x]6). (c) Is an isomorphism of rings? Explain. =
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![Denote by Zm = {[0]m, [1]m, ..., [m1]m} the ring of integers modulo m.
Consider the rings R = Z24 and SZ4 × Z6. Let 0 RS be the map
defined by ([x]24) = ([x]4, [4x]6).
(c) Is an isomorphism of rings? Explain.
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0de9773e-39c1-4df6-a7d6-864501c7f552%2F26f444c1-2cdd-4d89-bfe5-387bfbda9530%2Fxgf3nis_processed.png&w=3840&q=75)
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- 22. Let be a ring with finite number of elements. Show that the characteristic of divides .18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)
- Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y414. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove that is a ring homomorphism. Is (e)=e where e is the unity in 3 and e is the unity in 12?
- 14. Let be an ideal in a ring with unity . Prove that if then .Exercises If and are two ideals of the ring , prove that is an ideal of .11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .