Find the characteristic of the following rings.
d.
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Elements Of Modern Algebra
- Exercises If and are two ideals of the ring , prove that is an ideal of .arrow_forward19. Find a specific example of two elements and in a ring such that and .arrow_forwardTrue or false Label each of the following statements as either true or false. 7. For the quotient ring of by the ideal is .arrow_forward
- 14. Let be an ideal in a ring with unity . Prove that if then .arrow_forwardExercises If and are two ideals of the ring , prove that the set is an ideal of that contains each of and . The ideal is called the sum of ideals of and .arrow_forward15. Let and be elements of a ring. Prove that the equation has a unique solution.arrow_forward
- 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+y4arrow_forward17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.arrow_forwardTrue or False Label each of the following statements as either true or false. 3. The characteristic of a ring is zero if is the only integer such that for all in.arrow_forward
- 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_forwardLabel each of the following statements as either true or false. If I is an ideal of S where S is a subring of a ring R, then I is an ideal of R.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,