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Chapter 6 Solutions
Elements Of Modern Algebra
- a. If R is a commutative ring with unity, show that the characteristic of R[ x ] is the same as the characteristic of R. b. State the characteristic of Zn[ x ]. c. State the characteristic of Z[ x ].arrow_forward18. Let be a commutative ring with unity, and let be the principal ideal in . Prove that is isomorphic to .arrow_forward15. Let and be elements of a ring. Prove that the equation has a unique solution.arrow_forward
- 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?arrow_forward19. Find a specific example of two elements and in a ring such that and .arrow_forward37. Let and be elements in a ring. If is a zero divisor, prove that either or is a zero divisor.arrow_forward
- Exercises If and are two ideals of the ring , prove that is an ideal of .arrow_forwardLet R be a commutative ring with unity. Prove that deg(f(x)g(x))degf(x)+degg(x) for all nonzero f(x), g(x) in R[ x ], even if R in not an integral domain.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
- 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.)arrow_forwardExamples 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_forwardAssume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,