If
is an ideal of
Want to see the full answer?
Check out a sample textbook solutionChapter 6 Solutions
ELEMENTS OF MODERN ALGEBRA
- Exercises Find two ideals and of the ring such that is not an ideal of . is an ideal of .arrow_forwardLet I1 and I2 be ideals of the ring R. Prove that the set I1I2=a1b1+a1b2+...+anbnaiI1,biI2,nZ+ is an ideal of R. The ideal I1I2 is called the product of ideals I1 and I2.arrow_forward18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .arrow_forward
- 19. Find a specific example of two elements and in a ring such that and .arrow_forward14. Let be an ideal in a ring with unity . Prove that if then .arrow_forward24. 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_forward
- If R1 and R2 are subrings of the ring R, prove that R1R2 is a subring of R.arrow_forwardLabel each of the following statements as either true or false. The only ideal of a ring R that property contains a maximal ideal is the ideal R.arrow_forwardShow that the ideal is a maximal ideal of .arrow_forward
- Find the principal ideal (z) of Z such that each of the following sums as defined in Exercise 8 is equal to (z). (2)+(3) b. (4)+(6) c. (5)+(10) d. (a)+(b) If I1 and I2 are two ideals of the ring R, prove that the set I1+I2=x+yxI1,yI2 is an ideal of R that contains each of I1 and I2. The ideal I1+I2 is called the sum of ideals of I1 and I2.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_forwardTrue or false Label each of the following statements as either true or false. 3. The only ideal of a ring that contains the unity is the ring itself.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,