According to part
is a ring. Assume that the set
is an ideal of
Example
The set of all real numbers of the form
Want to see the full answer?
Check out a sample textbook solutionChapter 6 Solutions
Elements Of Modern Algebra
- 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 .)arrow_forward19. Find a specific example of two elements and in a ring such that and .arrow_forward[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]arrow_forward
- 15. Let and be elements of a ring. Prove that the equation has a unique solution.arrow_forward15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .arrow_forwardAn element in a ring is idempotent if . Prove that a division ring must contain exactly two idempotent e elements.arrow_forward
- Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an ideal of R.arrow_forward27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.arrow_forwardIf R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.arrow_forward
- Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here]arrow_forward32. a. Let be an ideal of the commutative ring and . Prove that the setis an ideal of containing . b. If and show that .arrow_forwardLet 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.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning