Answered: Theorem 15.2 Kernels Are Ideals Let o…

Name: Prove this theorem. Theorem 15.2 Kernels Are IdealsLet o be a ring hom
Uploaded: 2024-03-20T06:47:08.000Z
Channel: Bartleby
Description: Prove this theorem. Theorem 15.2 Kernels Are IdealsLet o be a ring homomorphism from a ring R to a ring S. Then Ker þ= {r ER\ 4(r) = 0} is an ideal of R.

Math

Algebra

Theorem 15.2 Kernels Are Ideals Let o be a ring homomorphism from a ring R to a ring S. Then Ker þ = {r ER\ 4(r) = 0} is an ideal of R.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.1: Ideals And Quotient Rings

Problem 7E: Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of

See similar textbooks

Similar questions

14. Let be an ideal in a ring with unity . Prove that if then .
Exercises 10. Prove Theorem 5.4:A subset of the ring is a subring of if and only if these conditions are satisfied: is nonempty. and imply that and are in .
17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.
Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of
Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.
24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)
15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .
32. a. Let be an ideal of the commutative ring and . Prove that the setis an ideal of containing . b. If and show that .
[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]
21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.
Exercises Prove Theorem 5.3:A subset S of the ring R is a subring of R if and only if these conditions are satisfied: S is nonempty. xS and yS imply that x+y and xy are in S. xS implies xS.
18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .